Single Equation Output

The Descriptive Statistics entry on the Model menu involves the formal calculation of statistics on database variables. Model-related statistics are considered in Chapters 17 and 18. This chapter provides the formulae underlying the computations. PcGive will use the largest available sample by default, here denoted by t=1,...,T. It is always possible to graph or compute the statistics over a shorter sample period.

16.1 Means, standard deviations and correlations

The correlation matrix of the selected variables is reported as a symmetric matrix with the diagonal equal to one. Each cell records the simple correlation between the two relevant variables. The same sample is used for each variable; observations with missing values are dropped.

16.2 Normality test and descriptive statistics

This is the test statistic described in §18.4.4, which amounts to testing whether the skewness and kurtosis of the variable corresponds to that of a normal distribution. Missing value are dropped from each variable, so the sample size may be different for each variable.

16.3 Autocorrelations (ACF) and Portmanteau statistic

This prints the sample autocorrelation function of the selected variables, as described in §18.4.2. The same sample is used for each variable; observations with missing values are dropped.

16.4 Unit-root tests

A crucial property of any economic variable influencing the behaviour of statistics in econometric models is the extent to which that variable is stationary. If the autoregressive description

has a root on the unit circle, then conventional distributional results are not applicable to coefficient estimates. As the simplest example, consider:

which generates a random walk (with drift if α≠0). Here, the autoregressive coefficient is unity and stationarity is violated. A process with no unit or explosive roots is said to be I(0); a process is I( d) if it needs to be differenced d times to become I(0) and is not I(0) if only differenced d-1 times. Many economic time series behave like I(1), though some appear to be I(0) and others I(2).

The Durbin--Watson statistic for the level of a variable offers one simple characterization of this integrated property:

If x_t is a random walk, DW will be very small. If x_t is white noise, DW will be around 2. Very low DW values thus indicate that a transformed model may be desirable, perhaps including a mixture of differenced and disequilibrium variables.

An augmented Dickey--Fuller (ADF) test for I(1) against I(0) (see Dickey and Fuller, 1981) is provided by the t-statistic on β̂ in:

The constant or trend can optionally be excluded from (eq:16.4); the specification of the lag length n assumes that u_t is white noise. The null hypothesis is H₀: β=0; rejection of this hypothesis implies that x_t is I(0). A failure to reject implies that Δx_t is stationary, so x_t is I(1). This is a second useful description of the degree of integratedness of x_t. The Dickey--Fuller (DF) test has no lagged first differences on the right-hand side ( n=0) . On this topic, see the Oxford Bulletin of Economics and Statistics (Hendry, 1986a, Banerjee and Hendry, 1992a), and Banerjee, Dolado, Galbraith and Hendry (1993). To test whether x_t is I(1), commence with the next higher difference:

Most of the formulae for the computed statistics are more conveniently presented in the next section on simple dynamic regressions, but the t-statistic is defined (e.g., for α̂) as t_α = α̂/SE(α̂), using the formula in (eq:17.5). Critical values are derived from the reponse surfaces in MacKinnon (1991), and depend on whether a constant, or constant and trend, are included (seasonals are ignored). Under the null (β=0), α≠0 entails a trend in {x_t} and μ≠0 implies a quadratic trend. However, under the stationary alternative, α=0 would impose a zero trend. Thus the test ceases to be similar if the polynomial in time (1,t,t² etc.) in the model is not at least as large as that in the data generating process (see, for example, Kiviet and Phillips, 1992). This problem suggests allowing for a trend in the model unless the data is anticipated to have a zero mean in differences. The so-called Engle-Granger two-step method amounts to applying the ADF test to residuals from a prior static regression (the first step). The response surfaces need to be adjusted for the number of variables involved in the first step: see MacKinnon (1991).

The default of PcGive is to report a summary test output for the sequence of ADF(n)...ADF(0) tests. The summary table lists, for j=n,...,0:

Critical values are given, and significance of the ADF test is marked by asterisks: ^* indicates significance at 5%, ^** at 1%.

16.5 Principal components analysis

Principal components analysis (PCA) amounts to an eigenvalue analysis of the correlation matrix. Because the correlation matrix has ones on the diagonal, its trace equals k when k variables are involved. Therefore, the sum of the eigenvalues also equals k. Moreover, all eigenvalues are non-negative.

where λ is the diagonal matrix with the ordered eigenvalues λ₁ ≥...≥λ_k ≥0 on the diagonal, and H=(h₁, ..., h_k) the matrix with the corresponding eigenvectors in the columns, H'H=I_k. The matrix of eigenvectors diagonalizes the correlation matrix:

Let (x₁, ..., x_k) denote the variables selected for principal components analysis (a T ×k matrix), and Z=(z₁, ..., z_k) the standardized data (i.e. in deviation from their mean, and scaled by the standard deviation). Then Z'Z/T = C. The jth principal component is defined as:

and accounts for 100 λ_j/k % of the variation. The largest m principal components together account for 100∑_j=1^m λ_j/k % of the variation.

Principal components analysis is used to capture the variability of the data in a small number of factors. Using the correlation matrix enforces a common scale on the data (analysis in terms of the variance matrix is not invariant to scaling). Some examples of the use of PCA in financial applications are given in Alexander (2001, Ch.6).

PCA is sometimes used to reconstruct missing data on y in combination with data condensation. Assume that T observations are available on y, but T+H on the remaining data, then two methods could be considered:

16.6 Correlogram, ACF

using the full sample mean x= 1/T ∑_t=1^Tx_t. The variance σ̂²_x corresponds to ĉ₀.

The autocorrelation function (ACF) plots the series {r̂_j} where r̂_j is the sample correlation coefficient between x_t and x_t-j. The length of the ACF is specified by the user, leading to a figure which shows ( r̂₁,r̂₂,...,r̂_s) plotted against ( 1,2,...,s) where for any j when x is any chosen variable:

The first autocorrelation, {r̂₀}, is equal to one, and omitted from the graphs.

The asymptotic variance of the autocorrelations is 1/T, so approximate 95% error bars are indicated at ±2T^-1/2 (see e.g. Harvey, 1993, p.42).

If a series is non-stationary, the usual definition of a correlation between successive lags is required: see Nielsen (2006a). This comment also applies to the partial autocorrelation function described in the next section.

16.7 Partial autocorrelation function (PACF)

Given the sample autocorrelation function {r̂_j}, the partial autocorrelations are computed using Durbin's method as described in Golub and Van Loan (1989, §4.7.2). This corresponds to recursively solving the Yule--Walker equations. For example, with autocorrelations, r̂₀, r̂₁, r̂₂, ..., the first partial correlation is α̂₀=1 (omitted from the graphs). The second, α̂₁, is the solution from

16.8 Periodogram

When the periodogram is plotted, only frequencies greater than zero and up to π are used. Moreover, the x-axis, with values 0,...,π, is represented as 0,...,1. So, when T=4 the x coordinates are 0.5,1 corresponding to π/2, π. When T=5, the x coordinates are 0.4,0.8 corresponding to 2π/5, 4π/5.

16.9 Spectral density

The estimated spectral density is a smoothed function of the sample autocorrelations {r̂_j}, defined as in (eq:16.7). The sample spectral density is then defined as:

where | ^.| takes the absolute value, so that, for example, r̂_{| -1|}=r̂₁. The K( ^.) function is called the lag window. OxMetrics uses the Parzen window:

We have that K(-j)=K(j), so that the sign of j does not matter ( cos (x)= cos (-x)). The r̂_js are based on fewer observations as j increases. The window function attaches decreasing weights to the autocorrelations, with zero weight for j>m. The parameter m is called the lag truncation parameter. In OxMetrics, this is taken to be the same as the chosen length of the correlogram. For example, selecting s=12 (the with length setting in the dialog) results in m=12. The larger m, the less smooth the spectrum becomes, but the lower the bias. The spectrum is evaluated at 128 points between 0 and π. For more information see Priestley (1981) and Granger and Newbold (1986).

16.10 Histogram, estimated density and distribution

Given a data set {x_t}=( x₁...x_T) which are observations on a random variable X. The range of {x_t} is divided into N intervals of length h with h defined below. Then the proportion of x_t in each interval constitutes the histogram; the sum of the proportions is unity on the scaling that is used. The density can be estimated as a smoothed function of the histogram using a normal or Gaussian kernel. This can then be summed (`integrated') to obtain the estimated cumulative distribution function (CDF).

Denote the actual density of X at x by f_x( x) . A non-parametric estimate of the density is obtained from the sample by:

where h is the window width or smoothing parameter, and K( ^.) is a kernel such that:

f_x( x) ̂ is usually calculated for 128 values of x, using a fast Fourier transform. An excellent reference on density function estimation is Silverman (1986).

16.11 QQ plot

The variable in a QQ plot would normally hold critical values which are hypothesized to come from a certain distribution. The QQ plot function then draws a cross plot of these observed values (sorted), against the theoretical quantiles. The 45^o line is drawn for reference (the closer the cross plot to this line, the better the match).

The normal QQ plot includes the pointwise asymptotic 95% standard error bands, as derived in Engler and Nielsen (2009)) for residuals of regression models (possibly autoregressive) with an intercept.

Chapter 17 Model Estimation Statistics

Once a model has been specified, a sample period selected, and an estimation method chosen, the equation can be estimated. OLS-CS/IVE-CS and OLS/IVE only differ in the way the sample period is selected. In the first, cross section, case, all observations with missing values are omitted. Therefore, `holes' in the database are simply skipped. In cross-section mode it is also possible to specify a variable Sel by which to select the sample. In that case, observations where Sel has a 0 or missing values are omitted from the estimation sample (but, if data is available, included in the prediction set). In dynamic regression, the observations must be consecutive in time, and the maximum available sample is the leading contiguous sample. The following table illustrates the default sample when regressing y on a constant:

For ease of notation, the sample period is denoted t=1,...,T+H, after allowing for any lagged variables created where H is the forecast horizon. The data used for estimation are X=( x₁...x_T) . The H retained observations X_H=( x_T+1...x_T+H) are used for static (1-step) forecasting and evaluating parameter constancy.

This chapter discusses the statistics reported by PcGive following model estimation. The next chapter presents the wide range of evaluation tools available following successful estimation. Sections marked with * denote information that can be shown or omitted on request. In the remainder there is no distinction between OLS/IVE and OLS-CS/IVE-CS.

17.1 Recursive estimation: RLS/RIVE/RNLS/RML

Recursive OLS and IV estimation methods are initialized by a direct estimation over t=1,...,M-1, followed by recursive estimation over t=M,...,T. RLS and RIVE update inverse moment matrices . This is inherently somewhat numerically unstable, but, because it is primarily a graphical tool, this is not so important.

Recursive estimation of non-linear models is achieved by the brute-force method: first estimate for the full sample, then shrink the sample by one observation at a time. At each step the estimated parameters of the previous step are used as starting values, resulting in a considerably faster algorithm.

The final estimation results are always based on direct full-sample estimation, so unaffected whether recursive or non-recursive estimation is used. The recursive output can be plotted from the recursive graphics dialog.

17.2 OLS estimation

The algebra of OLS estimation is well established from previous chapters. The model is:

Forecast statistics are provided for the H retained observations (only if H≠0). For OLS, these are comprehensive 1-step ahead forecasts and tests, described below.

The estimation output is presented in columnar format, where each row lists information pertaining to each variable (its coefficient, standard error, t-value, etc.). Optionally, the estimation results can be printed in equation format,, which is of the form coefficient × variable with standard errors in parentheses underneath.

17.2.1 The estimated regression equation

The first column of these results records the names of the variables and the second, the estimated regression coefficients β̂=( X'X) ^-1X'y. PcGive does actually not use this expression to estimate β̂. Instead it uses the QR decomposition with partial pivoting, which analytically gives the same result, but in practice is a bit more reliable (i.e. numerically more stable). The QR decomposition of X is X=QR, where Q is T ×T and orthogonal (that is, Q'Q=I), and R is T ×k and upper triangular. Then X'X=R'R.

The following five columns give further information about each of the magnitudes described below in §17.2.2 to §17.2.11.

17.2.2 Standard errors of the regression coefficients

where d_ii is the i^th diagonal element of ( X'X) ^-1 and σ̂_u is the standard error of the regression, defined in (eq:17.4).

17.2.3 t-values and t-probability

These statistics are conventionally calculated to determine whether individual coefficients are significantly different from zero:

where the null hypothesis H₀ is β_i=0. The null hypothesis is rejected if the probability of getting a t-value at least as large is less than 5% (or any other chosen significance level). This probability is given as:

in which τ has a Student t-distribution with T-k degrees of freedom. The t-probabilities do not appear when all other options are switched on.

When H₀ is true (and the model is otherwise correctly specified in a stationary process), a Student t-distribution is used since the sample size is often small, and we only have an estimate of the parameter's standard error: however, as the sample size increases, τ tends to a standard normal distribution under H₀. Large t-values reject H₀; but, in many situations, H₀ may be of little interest to test. Also, selecting variables in a model according to their t-values implies that the usual (Neyman--Pearson) justification for testing is not valid (see, for example, Judge, Griffiths, Hill, Lütkepohl and Lee, 1985).

17.2.4 Squared partial correlations

The final column lists the squared partial correlations under the header Part.R^2. The j^th entry in this column records the correlation of the j^th explanatory variable with the dependent variable, given the other k-1 variables. Adding further explanatory variables to the model may either increase or lower the squared partial correlation, and the former may occur even if the added variables are correlated with the already included variables. If the squared partial correlations fall on adding a variable, then that is suggestive of collinearity for the given equation parametrization: that is, the new variable is a substitute for, rather than a complement to, those already included.

Beneath the columnar presentation an array of summary statistics is also provided as follows:

17.2.5 Equation standard error (̂σ)

17.2.6 Residual sum of squares (RSS)

17.2.7 R²: squared multiple correlation coefficient

The variation in the dependent variable, or the total sum of squares (TSS), can be broken up into two parts: the explained sum of squares (ESS) and the residual sum of squares (RSS). In symbols, TSS=ESS+RSS, or:

assuming a constant is included. Thus, R² is the proportion of the variance of the dependent variable which is explained by the variables in the regression. By adding more variables to a regression, R² will never decrease, and it may increase even if nonsense variables are added. Hence, R² may be misleading. Also, R² is dependent on the choice of transformation of the dependent variable (for example, y versus Δy) -- as is the F-statistic below. The equation standard error, σ̂_u, however, provides a better comparative statistic because it is adjusted by the degrees of freedom. Generally, σ̂ can be standardized as a percentage of the mean of the original level of the dependent variable (except if the initial mean is zero) for comparisons across specifications. Since many economic magnitudes are inherently positive, that standardization is often feasible. If y is in logs, 100σ̂ is the percentage standard error.

17.2.8 F-statistic

Here, the null hypothesis is that the population R² is zero, or that all the regression coefficients are zero (excluding the intercept). The value for the F-statistic is followed by its probability value between square brackets.

17.2.9 R²: Adjusted R²

assuming a constant is included. The adjusted R-squared can go down when the number of variables increases. Nonetheless, there is no rationale to use it as a model selection criterion.

17.2.10 Log-likelihood

17.2.11 Mean and standard error of dependent variable

The final entries list the number of observations used in the regressor (so after allowing for lags), and the number of estimated parameters. This is followed by the mean and standard error of the dependent variable:

Note that we use T-1 in the denominator of σ̂²_y, so this is what would be reported as the equation standard error (eq:17.10) when regressing the dependent variable on just a constant.

17.2.12 *Information criteria

The four statistics reported are the Schwarz criterion (SC), the Hannan--Quinn (HQ) criterion, the Final Prediction Error (FPE), and the Akaike criterion (AIC). Here:

17.2.13 *Heteroscedastic-consistent standard errors (HCSEs)

These provide consistent estimates of the regression coefficients' standard errors even if the residuals are heteroscedastic in an unknown way. Large differences between the corresponding values in §17.2.2 and §17.2.13 are indicative of the presence of heteroscedasticity, in which case §17.2.13 provides the more useful measure of the standard errors (see White, 1980). PcGive contains two methods of computing heteroscedastic-consistent standard errors: as described in White (1980) (labelled HCSE), or the Jack-knife estimator from MacKinnon and White (1985) (labelled JHCSE; for which the code was initially provided by James MacKinnon).

The heteroscedasticity and autocorrelation consistent standard errors are reported in the column labelled HACSE. This follows Newey and West (1987), also see Andrews (1991).

17.2.14 *R² relative to difference and seasonals

The R² is preceded by the seasonal means s of the first difference of the dependent variable (\Delta y for annual data, four quarterly means for quarterly data, twelve monthly means for monthly data etc.).

The R² relative to difference and seasonals is a measure of the goodness of fit relative to ∑(Δy_t-s)² instead of ∑(y_t-y)² in the denominator of R² (keeping ∑û_t² in the numerator), where s denotes the relevant seasonal mean. Despite its label, such a measure can be negative: if it is, the fitted model does less well than a regression of Δy_t on seasonal dummies.

17.2.15 *Correlation matrix of regressors

This reports the sample means and sample standard deviations of the selected variables:

The correlation matrix of the selected variables is reported as a lower-triangular matrix with the diagonal equal to one. Each cell records the simple correlation between the two relevant variables. The calculation of the correlation coefficient r_xy between x and y is:

17.2.16 *Covariance matrix of estimated parameters

The matrix of the estimated parameters' variances is reported as lower triangular. Along the diagonal, we have the variance of each estimated coefficient, and off the diagonal, the covariances. The k×k variance matrix of β̂ is estimated by:

where σ̂² is the full-sample equation error variance. The variance-covariance matrix is only shown when requested, in which case it is reported before the equation output.

The remaining statistics only appear if observations were withheld for forecasting purposes:

17.2.17 1-step (ex post) forecast analysis

which requires the observations X_H'=(x_T+1,...,x_T+H). The 1-step forecast error is the mistake made each period:

This corresponds to the results given for the innovations in recursive estimation. The whole vector of forecast errors is e=( e_T+1,...,e_T+H) '. V[e] is derived in a similar way:

The columns respectively report the date for which the forecast is made, the realized outcome (y_t), the forecast (ŷ_t), the forecast error (e_t=y_t-ŷ_t), the standard error of the 1-step forecast (SE( e_t) =√V[ e_t] ̂), and a t-value (that is, the standardized forecast error e_t/SE( e_t) ).

17.2.18 Forecast test

A χ² statistic follows the 1-step analysis, comparing within and post-sample residual variances. Neither this statistic nor η₃ below measure absolute forecast accuracy. The statistic is calculated as follows:

The null hypothesis is `no structural change in any parameter between the sample and the forecast periods' (denoted 1 and 2 respectively), H₀: β₁=β₂; σ₁²=σ₂². A rejection of the null hypothesis of constancy by ξ₃ below implies a rejection of the model used over the sample period -- so that is a model specification test -- whereas the use of ξ₁ is more as a measure of numerical parameter constancy, and it should not be used as a model-selection device (see Kiviet, 1986). However, persistently large values for this statistic imply that the equation under study will not provide very accurate ex ante predictions, even one step ahead. An approximate F-equivalent is given by:

A second statistic takes parameter uncertainty into account, taking the denominator from (eq:17.20):

This test is not reported in single-equation modelling, but individual terms of the summation can be plotted in the graphical analysis.

17.2.19 Chow test

where H₀ is as for ξ₁. For fixed regressors, the Chow (1960) test is exactly distributed as an F, but is only approximately (or asymptotically) so in dynamic models.

We can now see the relation between ξ₃ and ξ₁: the latter uses V[ e] ̂=σ̂_u²I, obtained by dropping the (asymptotically negligible) term V[β̂] in (eq:17.21). In small samples, the dropped term is often not negligible, so ξ₁ should not be taken as a test. The numerical value of ξ₁ always exceeds that of ξ₃: the difference indicates the relative increase in prediction uncertainty arising from estimating, rather than knowing, the parameters.

17.2.20 t-test for zero forecast innovation mean (RLS only)

The recursive formulae are applicable over the sample T+1,...,T+H, and under the null of correct specification and H₀ of ξ₁ above, then the standardized innovations {ν_t/(ω_t)^1/2} are distributed as IN(0,σ²_u). Thus:

This tests for a different facet of forecast inaccuracy in which the forecast errors have a small but systematic bias. This test is the same as an endpoint CUSUM test of recursive residuals, but using only the forecast sample period (see Harvey and Collier, 1977).

17.3 IV estimation

in which we have n-1 endogenous variables y_t^* and q₁ non-modelled variables w_t on the right-hand side (the latter may include lagged endogenous variables). We assume that we have q₂ additional instruments, labelled w_t^*. Write y_t=(y_t:y_t^*')' for the n×1 vector of endogenous variables. Let z_t denote the set of all instrumental variables (non-endogenous included regressors, plus additional instruments): z_t=(w_t':w_t^*')', which is a vector of length q=q₁+q₂.

17.3.1 *Reduced form estimates

The reduced form (RF) estimates are only printed on request. If Z^'=( z₁...z_T) , and y_t denotes all the n endogenous variables including y_t at t with Y'=(y₁,...,y_T), then the RF estimates are:

which is q×n. The elements of Π̂' relevant to each endogenous variable are written:

with Y_i'=(y_i1,...,y_iT) the vector of observations on the i^th endogenous variable. Standard errors etc. all follow as for OLS above (using Z, Y_i for X,y in the relevant equations there).

17.3.2 Structural estimates

Generalized instrumental variables estimates for the k=n-1+q₁ coefficients of interest β=(β₀':β₁')' are:

using x_t=(y_t^*':w_t')', X'=( x₁...x_T) , y=(y₁...y_T)', which is the left-hand side of (eq:17.29), and Z is as in (eq:17.30). This allows for the case of more instruments than explanatory variables (q>k), and requires rank(X'Z)=k and rank(Z'Z)=q. If q=k the equation simplifies to:

As for OLS, PcGive does not use expression (eq:17.32) directly, but instead uses the QR decomposition for numerically more stable computation. The error variance is given by

Again the output is closely related to that reported for least squares except that the columns for HCSE, partial r² and instability statistics are omitted. However, RSS, σ̃ and DW are recorded, as is the reduced form σ̂ (from regressing y_t on z_t, already reported with the RF equation for y_t). Additional statistics reported are :

17.3.3 Specification χ²

This tests for the validity of the choice of the instrumental variables as discussed by Sargan (1964). It is asymptotically distributed as χ²(q₂-n+1) when the q₂-n+1 over-identifying instruments are independent of the equation error. It is also interpretable as a test of whether the restricted reduced form (RRF) of the structural model (y_t on x_t plus x_t on z_t) parsimoniously encompasses the unrestricted reduced form (URF: y_t on z_t directly):

17.3.4 Testing β=0

Reported is the χ² test of β=0 (other than the intercept) which has a crude correspondence to the earlier F-test. On H₀: β=0, the reported statistic behaves asymptotically as a χ²( k-1) . First define

Then ξ_β /σ̃_ε _app ̃ χ²(k) would test whether all k coefficients are zero. To keep the intercept separate, we compute:

This amounts to using the formula for β̃ (eq. (eq:17.32)) in ξ_β with y-yι instead of y.

17.3.5 Forecast test

A forecast test is provided if H observations are retained for forecasting. For IVE there are endogenous regressor variables: the only interesting issue is that of parameter constancy and correspondingly the output is merely ξ₁ of (eq:17.22) using σ̃_ε and:

Dynamic forecasts (which require forecasts of the successive x_T+1,...,x_T+H) could be obtained from multiple equation dynamic modelling, where the system as a whole is analyzed.

17.4 RALS estimation

As discussed in the typology, if a dynamic model has common factors in its lag polynomials, then it can be re-expressed as having lower-order systematic dynamics combined with an autoregressive error process (called COMFAC. If the autoregressive error is of r^th order, the estimator is called r^th-order Autoregressive Least Squares or RALS, and it takes the form:

as a function of the ( β,α) parameters yields a non-linear least squares problem necessitating iterative solution. However, conditional on values of either set of parameters, f( ^.) is linear in the other set, so analytical first and second derivatives are easy to obtain. There is an estimator-generating equation for this whole class (see Hendry, 1976, Section 7), but as it has almost no efficient non-iterative solutions, little is gained by its exploitation. Letting θ denote all of the unrestricted parameters in β₀( ^.) , {β_i( ^.) } and α( ^.) , then the algorithm programmed in PcGive for maximizing f( ^.) as a function of θ is a variant of the Gauss--Newton class. Let:

where s_i is a scalar chosen by a line search procedure to maximize f(θ_i+1|θ_i). The convergence criterion depends on q_i'Q_i^-1q_i and on changes in θ_i between iterations. The bi-linearity of f( ^.) is exploited in computing Q.

17.4.1 Initial values for RALS

Before estimating by RALS, OLS estimates of {β_i} are calculated, as are LM-test values of {α_i}, where the prespecified autocorrelation order is `data frequency+1' (for example, 5 for quarterly data). These estimates are then used to initialize θ. However, the {α_i} can be reset by users. Specifically, for single-order processes, u_t=α_ru_t-r+ε_t, then α_r can be selected by a prior grid search. The user can specify the maximum number of iterations, the convergence tolerance, both the starting and ending orders of the polynomial α( L) in the form:

and whether to minimize f( ^.) sequentially over s, s+1,...,r or merely the highest order, r.

17.4.2 Final estimates

On convergence, the variances of the θs are calculated (from Q^-1), as are the roots of α( L) =0. The usual statistics for σ̂, RSS (this can be used in likelihood-ratio tests between alternative nested versions of a model), t-values etc. are reported, as is T^-1∑( y_t-y) ² in case a pseudo-R² statistic is desired.

17.4.3 Analysis of 1-step forecasts

where we define x_t^+'=x_t-Σ_s^rα_ix_t-i, û_r'=( û_t-s...û_t-r) , w_t'=( x_t^+':û_r') , and θ'=( β':α') when α^'=( α_s...α_r) . E[ e_t] ~=0 for a correctly-specified model. Finally, therefore (neglecting the second-order dependence of the variance of w_t'(θ-θ̂) on θ̂ acting through w_t):

V[θ̂] is the RALS variance-covariance matrix, and from the forecast-error covariance matrix, the 1-step analysis is calculated, as are parameter-constancy tests.

The output is as for OLS: the columns respectively report the date for which the forecast is made, the realized outcome (y_t), the forecast (ŷ_t), the forecast error (e_t=y_t-ŷ_t), the standard error of the 1-step forecast (SE( e_t) =√V[ e_t] ̂), and a t-value (that is, the standardized forecast error e_t/SE( e_t) ).

17.4.4 Forecast tests

The RALS analogues of the forecast test ξ₁ of (eq:17.22), and of the Chow test η₃ in (eq:17.26), are reported. The formulae follow directly from (eq:17.48) and (eq:17.53).

17.5 Non-linear modelling

17.5.1 Non-linear least squares (NLS) estimation

Note that for fixed θ₂ this last model becomes linear; for example, for θ₂= 1/2 :

which is linear in the transformed variables x_t^*, z_t^*. As for OLS, estimation proceeds by minimizing the sum of squared residuals:

In linear models, this problem has an explicit solution; for non-linear models the minimum has to be found using iterative optimization methods.

Instead of minimizing the sum of squares, PcGive maximizes the sum of squares divided by -T:

with q( ^.) the derivatives of g(^.) with respect to θ_j (this is determined numerically), and Q( ^.) ^-1 a symmetric, positive definite matrix (determined by the BFGS method after some initial Gauss-Newton steps). Practical details of the algorithm are provided in §17.5.3; Volume II gives a more thorough discussion of the subject of numerical optimization. Before using NLS you are advised to study the examples given in the tutorial Chapter, to learn about the potential problems.

Output is as for OLS, except for the instability tests and HCSEs which are not computed. The variance of the estimated coefficients is determined numerically, other statistics follow directly, for example:

Forecasts are computed and graphed, but the only statistic reported is the ξ₁ test of (eq:17.22), using 1-step forecast errors:

17.5.2 Maximum likelihood (ML) estimation

Models falling in this class are, for example, binary logit and probit, ARCH, GARCH, Tobit, Poisson regression. As an example, consider the linear regression model. PcGive gives three ways of solving this:

Estimation of (eq:17.61) uses the same technique as NLS. The output is more concise, consisting of coefficients, standard errors (based on the numerical second derivative), t-values, t-probabilities, and `loglik' which is ∑_t=1^Tl(θ̂|x_t). Forecasts are computed and graphed, but no statistics are reported.

17.5.3 Practical details

Non-linear model are formulated in algebra code. NLS requires the definition of a variable called actual, and one called fitted. It uses these to maximize minus the residual sum of squares divided by T:

This is just a linear model, and much more efficiently done using the normal options.

Models can be estimated by maximum likelihood if they can be written as a sum over the observations (note that the previous concentrated log-likehood cannot be written that way!). An additional algebra line is required, to define a variable called loglik. PcGive maximizes:

Here actual and fitted are not really that, but these variables define what is being graphed in the graphic analysis.

Note that algebra is a vector language without temporary variables, restricting the class of models that can be estimated. Non-linear models are not stored for recall and progress reports.

After correct model specification, the method is automatically set to Non-linear model (using ML if loglik is defined, NLS/RNLS otherwise); in addition, the following information needs to be specified:

NLS and ML estimation (and their recursive variants RNLS and RML) require numerical optimization to maximize the likelihood log L( φ( θ) ) = l( φ( θ) ) as a non-linear function of θ. PcGive maximization algorithms are based on a Newton scheme:

PcGive uses the quasi-Newton method developed by Broyden, Fletcher, Goldfarb, Shanno (BFGS) to update K = Q^-1 directly. It uses numerical derivatives to compute ∂l( φ( θ) ) / ∂θ_i. However, for NLS, PcGive will try Gauss-Newton before starting BFGS. In this hybrid method, Gauss-Newton is used while the relative progress in the function value is 20%, then the program switches to BFGS.

Starting values must be supplied. The starting value for K consistes of 0s off-diagonal. The diagonal is the minimum of one and the inverse of the corresponding diagonal element in the matrix consisting of the sums of the outer-products of the gradient at the parameter starting values (numerically evaluated).

RNLS works as follows: starting values for θ and K for the first estimation (T-1 observations) are the full sample values (T observations); then the sample size is reduced by one observation; the previous values at convergence are used to start with.

Owing to numerical problems it is possible (especially close to the maximum) that the calculated δ_i does not yield a higher likelihood. Then an s_i∈[0,1] yielding a higher function value is determined by a line search. Theoretically, since the direction is upward, such an s_i should exist; however, numerically it might be impossible to find one. When using BFGS with numerical derivatives, it often pays to scale the data so that the initial gradients are of the same order of magnitude.

The convergence decision is based on two tests. The first uses likelihood elasticities (∂l/∂ log θ):

The second is based on the one-step-ahead relative change in the parameter values:

Chapter 18 Model Evaluation Statistics

18.1 Graphic analysis

Graphic analysis focuses on graphical inspection of individual equations. Let y_t, ŷ_t denote respectively the actual (that is, observed) values and the fitted values of the selected equation, with residuals û_t=y_t-ŷ_t, t=1,...,T. When H observations are retained for forecasting, then ŷ_T+1,...,ŷ_T+H are the 1-step forecasts. NLS/RNLS/ML use the variables labelled `actual' and `fitted' for y_t, ŷ_t.

18.2 Recursive graphics (RLS/RIVE/RNLS/RML)

Recursive methods estimate the model at each t for t=M-1,...,T. The output generated by the recursive procedures is most easily studied graphically, possibly using the facility to view multiple graphs together on screen. The dialog has a facility to write the output to the editor, instead of graphing it. The recursive estimation aims to throw light on the relative future information aspect (that is, parameter constancy).

Let β̂_t denote the k parameters estimated from a sample of size t, and y_j-x_j'β̂_t the residuals at time j evaluated at the parameter estimates based on the sample 1,...,t (for RNLS the residuals are y_j-f(x_j,β̂_t)).

The statistics in (eq:18.1)--(eq:18.3) are variants of Chow (1960) tests: they are scaled by 1-off critical values from the F-distribution at any selected probability level as an adjustment for changing degrees of freedom, so that the significant critical values become a straight line at unity. Note that the first and last values of (eq:18.1) respectively equal the first value of (eq:18.3) and the last value of (eq:18.2).

The Chow test statistics are not calculated for RIVE/RML; the recursive RSS is not available for RML.

18.3 Dynamic analysis

where b₀( L) and the b_i( L) are polynomials in the lag operator L. Now q+1 is the number of distinct variables (one of which is y_t), whereas k remains the number of estimated coefficients. For simplicity we take all polynomials to be of length m:

In its unrestricted mode of operation, PcGive can be visualized as analyzing the polynomials involved, and it computes such functions as their roots and sums. This option is available if a general model was initially formulated, and provided OLS or IVE was selected.

18.3.1 Static long-run solution

When working with dynamic models, concepts such as equilibrium solutions, steady-state growth paths, mean lags of response etc. are generally of interest. In the simple model:

in which case, E[ y_t] =y^* will also be constant if |α₁|<1, and y_t will converge to:

For non-stationary but cointegrated data, reinterpret expression (eq:18.7) as E[ y_t-Kz_t] =0.

PcGive computes estimates of K and associated standard errors. These are called static long-run parameters. If b₀( 1) ≠0, the general long-run solution of (eq:18.4) is given by:

The expression y_t-ΣK_iz_it is called the equilibrium-correction mechanism (ECM) and can be stored in the data set. If common-factor restrictions of the form b_j( L) =α( L) γ_j( L) , j=0,...,q are imposed, then α( 1) will cancel, hence enforced autoregressive error representations have no impact on derived long-run solutions.

PcGive calculates J analytically using the algorithm proposed by Båardsen (1989).

PcGive outputs the solved static long-run equation, with standard errors of the coefficients. This is followed by a Wald test of the null that all of the long-run coefficients are zero (except the constant term). The V[K̂]̂ matrix is printed when `covariance matrix of estimated coefficients' is checked under the model options.

18.3.2 Analysis of lag structure

The b̂_i( L) , i=0,...,q of (eq:18.4) and their standard errors are reported in tabular form with the b̂_i( 1) (their row sums) and associated standard errors.

18.3.2.1 Tests on the significance of each variable

These test the significance of each basic variable in turn. The final column gives the PcGive unit-root tests:

If H_ui: b_i( 1) =0 cannot be rejected, there is no significant long-run level effect from z_it; if H_vi: b_i=0 cannot be rejected, there is no significant effect from z_it at any (included) lag. Significance is marked by ^* for 5% and ^** for 1%. Critical values for the PcGive unit-root test (H_u0: b₀( 1) =0) are based on Ericsson and MacKinnon (2002). For the unit-root test, only significance of the dependent variable is reported (not the remaining variables!),

Note that b_i( 1) =0 and b_i=0 are not equivalent; testing K_i=0 is different again. Using (eq:18.6) we can show the relevant hypotheses:

18.3.2.2 Tests on the significance of each lag

F-tests of each lag length are shown, beginning at the longest ( m) and continuing down to 1. The test of the longest lag is conditional on keeping lags ( 1,...,m-1) , that of ( m-1) is conditional on ( 1,...,m-2,m) etc.

18.3.3 Tests on the significance of all lags

Finally, F-tests of all lags up to m are shown, beginning at the longest ( 1,...,m) and continuing further from ( 2,...,m) down to ( m,...,m) . These tests are conditional on keeping no lags, keeping lag 1, down to keeping ( 1,...,m-1) . Thus, they show the marginal significance of all longer lags.

18.3.4 COMFAC tests

where α( L) is of order r and ^* denotes polynomials of the original order minus r. The degrees of freedom for the Wald tests for COMFAC are equal to the number of restrictions imposed by α( L) and the Wald statistics are asymptotically χ² with these degrees of freedom if the COMFAC restrictions are valid. It is preferable to use the incremental values obtained by subtracting successive values of the Wald tests. These are χ² also, with degrees of freedom given by the number of additional criteria. Failure to reject common-factor restrictions does not entail that such restrictions must be imposed. For a discussion of the theory of COMFAC, see Hendry and Mizon (1978) for some finite-sample Monte Carlo evidence see Mizon and Hendry (1980). COMFAC is not available for RALS.

When the minimum order of lag length in the b_i( L) is unity or larger (m say), the Wald test sequence for 1,2,...,m common factors is calculated. Variables that are redundant when lagged (Constant, Seasonals, Trend) are excluded in conducting the Wald test sequence since they always sustain a common-factor interpretation.

18.3.5 Lag weights

Starting from an equilibrium z^* at t=0, a one-off increment of δ to z^* has an impact on y^* at t=0,1,2,... of w₀δ, w₁δ, w₂δ, w₃δ,... with the ws defined by equating coefficients of powers of L as:

PcGive can graph the normalized lag weights w₀/w( 1) , w₁/w( 1) ,..., w_s/w( 1) and the cumulative normalized lag weights w₀/w( 1) , ( w₀+w₁) /w( 1) ,..., ( w₀+...+w_s) /w( 1) .

18.4 Diagnostic tests

18.4.1 Introduction

Irrespective of the estimator selected, a wide range of diagnostic tests is offered. Tests are available for residual autocorrelation, conditional heteroscedasticity, normality, unconditional heteroscedasticity/functional form mis-specification and omitted variables. Recursive residuals can be used if these are available. Tests for common factors and linear restrictions are discussed in §18.3.4 and §18.5 below, encompassing tests in §18.9. Thus, relating this section to the earlier information taxonomy , the diagnostic tests of this section concern the past (checking that the errors are a homoscedastic, normal, innovation process relative to the information available), whereas the forecast statistics discussed in Chapter 17 concern the future and encompassing tests concern information specific to rival models.

Many test statistics in PcGive have either a χ² distribution or an F distribution. F-tests are usually reported as:

where the test statistic has an F-distribution with one degree of freedom in the numerator, and 155 in the denominator. The observed value is 5.0088, and the probability of getting a value of 5.0088 or larger under this distribution is 0.0266. This is less than 5% but more than 1%, hence the star. Significant outcomes at a 1% level are shown by two stars. χ² tests are also reported with probabilities, as for example:

The 5% χ² critical values with two degrees of freedom is 5.99, so here normality is not rejected (alternatively, Prob(χ²≥ 2.1867) = 0.3351, which is more than 5%). Details on the computation of probability values and quantiles for the F and χ² tests are given under the probf, probchi, quanf and quanchi functions in the Ox reference manual (Doornik, 2007b).

Some tests take the form of a likelihood ratio (LR) test. If l is the unrestricted, and l₀ the restricted log-likelihood, then -2(l₀-l) has a χ²(s) distribution, with s the number of restrictions imposed (so model l₀ is nested in l).

Many diagnostic tests are calculated through an auxiliary regression. For single-equation tests, they take the form of TR² for the auxiliary regression so that they are asymptotically distributed as χ²( s) under their nulls, and hence have the usual additive property for independent χ²s. In addition, following Harvey (1990) and Kiviet (1986), F-approximations are calculated because they may be better behaved in small samples:

When the covariance matrix is block diagonal between regression and heteroscedasticity (or ARCH) function parameters, tests can take the regression parameters as given, see Davidson and MacKinnon (1993, Ch. 11):

This may be slightly different if not all parameters are included in the test, or when observations are lost in the construction of the test.

18.4.2 Residual autocorrelations (ACF), Portmanteau and DW

The sample autocorrelation function (ACF) of a variable x_t is the series {r_j} where r_j is the correlation coefficient between x_t and x_t-j for j = 1,...,s:

The residual correlogram is defined as above, but using the residuals from the econometric regression, rather than the data. Thus, this reports the series {r_j} of correlations between the residuals û_t and û_t-j. In addition, PcGive prints the partial autocorrelation function (PACF) (see the OxMetrics book).

It is possible to calculate a statistic based on `T*(sum of s squared autocorrelations)', with s the length of the correlogram, called the Portmanteau statistic:

This is corresponds to Box and Pierce (1970), but with a degrees of freedom correction as suggested by Ljung and Box (1978). It is designed as a goodness-of-fit test in stationary, autoregressive moving-average models. Under the assumptions of the test, LB(s) is asymptotically distributed as χ²(s-n) after fitting an AR(n) model. A value such that LB( s) ≥2s is taken as indicative of mis-specification for large s. However, small values of such a statistic should be treated with caution since residual autocorrelations are biased towards zero (like DW) when lagged dependent variables are included in econometric equations. An appropriate test for residual autocorrelation is provided by the LM test in §18.4.3 below.

18.4.2.1 Durbin--Watson statistic (DW)

If 0<DW<2, then the null hypothesis is H₀: ρ=0, that is, zero autocorrelation (so DW=2) and the alternative is H₁: ρ>0, that is, positive first-order autocorrelation.

If 2<DW<4, then H₀: ρ=0 and H₁: ρ<0, in which case DW^*=4-DW should be computed.

The significance values of DW are widely recorded in econometrics' textbooks. However, DW is a valid statistic only if all the x_t variables are non-stochastic, or at least strongly exogenous. If the model includes a lagged dependent variable, then DW is biased towards 2, that is, towards not detecting autocorrelation, and Durbin's h-test (see Durbin, 1970) or the equivalent LM-test for autocorrelation in §18.4.3 should be used instead. Also see §16.4.

18.4.3 Error autocorrelation test (not for RALS, ML)

This is the Lagrange-multiplier test for r^th order residual autocorrelation, distributed as χ²( r) in large samples, under the null hypothesis that there is no autocorrelation (that is, that the errors are white noise). In standard usage, r~= 1/2 s for s in §18.4.2 above, so this provides a type of Portmanteau test (see Godfrey, 1978). However, any orders from 1 up to 12 can be selected to test against:

As noted above, the F-form suggested by Harvey (1981, see Harvey, 1990) is the recommended diagnostic test. Following the outcome of the F-test (and its p-value), the error autocorrelation coefficients are recorded. For an autoregressive error of order r to be estimated by RALS, these LM coefficients provide good initial values, from which the iterative optimization can be commenced. The LM test is calculated by regressing the residuals on all the regressors of the original model and the lagged residuals for lags p to r (missing residuals are set to zero). The LM test χ²(r-p+1) is TR² from this regression (or the F-equivalent), and the error autocorrelation coefficients are the coefficients of the lagged residuals. For an excellent exposition, see Pagan (1984).

18.4.4 Normality test

Let μ, σ_x² denote the mean and variance of {x_t}, and write μ_i=E[ x_t-μ] ⁱ, so that σ_x²=μ₂. The skewness and kurtosis are defined as:

which subsequently was derived as an LM test by Jarque and Bera (1987). Unfortunately e₁ has rather poor small sample properties: √b₁ and b₂ are not independently distributed, and the sample kurtosis especially approaches normality very slowly. The test reported by PcGive is based on Doornik and Hansen (1994), who employ a small sample correction, and adapt the test for the multivariate case. It derives from Shenton and Bowman (1977), who give b₂ (conditional on b₂>1+b₁) a gamma distribution, and D'Agostino (1970), who approximates the distribution of √b₁ by the Johnson S_u system. Let z₁ and z₂ denote the transformed skewness and kurtosis, where the transformation creates statistics which are much closer to standard normal. The test statistic is:

Table Table:18.1 compares (eq:18.19) with its asymptotic form (eq:18.18). It gives the rejection frequencies under the null of normality, using χ²(2) critical values. The experiments are based on 10000 replications and common random numbers.

Table:18.1 Empirical size of normality tests

PcGive reports the following statistics under the normality test option, replacing x_t by the residuals u_t:

18.4.5 Heteroscedasticity test using squares (not for ML)

This test is based on White (1980), and involves an auxiliary regression of {û_t²} on the original regressors ( x_it) and all their squares (x_it²). The null is unconditional homoscedasticity, and the alternative is that the variance of the {u_t} process depends on x_t and on the x_it². The output comprises TR², the F-test equivalent, the coefficients of the auxiliary regression, and their individual t-statistics, to help highlight problem variables. Variables that are redundant when squared are automatically removed, as are observations that have a residual that is (almost) zero. Some additional information can be found in Volume II.

18.4.6 Heteroscedasticity test using squares and cross-products (not for ML)

This test is that of White (1980), and only calculated if there is a large number of observations relative to the number of variables in the regression. It is based on an auxiliary regression of the squared residuals ( û_t²) on all squares and cross-products of the original regressors (that is, on r=½k( k+1) variables). That is, if T>>k( k+1) , the test is calculated; redundant variables are automatically removed, as are observations that have a residual that is (almost) zero. The usual χ² and F-values are reported; coefficients of the auxiliary regression are also shown with their t-statistics to help with model respecification. This is a general test for heteroscedastic errors: H₀ is that the errors are homoscedastic or, if heteroscedasticity is present, it is unrelated to the xs.

In previous versions of PcGive this test used to be called a test for functional form mis-specification. That terminology was criticized by Godfrey and Orme (1994), who show that the test does not have power against omitted variables.

18.4.7 ARCH test

This is the ARCH (AutoRegressive Conditional Heteroscedasticity) test: see Engle, 1982) which in the present form tests the hypothesis γ=0 in the model:

where γ=( γ₁,...,γ_r) '. Again, we have TR² as the χ² test from the regression of û_t² on a constant and û_t-1² to û_t-r² (called the ARCH test) which is asymptotically distributed as χ²( r) on H₀: γ=0. The F-form is also reported. Both first-order and higher-order lag forms are easily calculated (see Engle, 1982, and Engle, Hendry and Trumbull, 1985).

18.4.8 RESET (OLS only)

The RESET test (Regression Specification Test) due to Ramsey (1969) tests the null of correct specification of the original model against the alternative that powers of ŷ_t such as (ŷ_t², ŷ_t³...) have been omitted (PcGive only allows squares). This tests to see if the original functional form is incorrect, by adding powers of linear combinations of xs since by construction, ŷ_t=x_t^'β̂_t.

We use RESET23 for the test that uses squares and cubes, while RESET refers to the test just using squares.

18.4.9 Parameter instability tests (OLS only)

Parameter instability statistics are reported for σ², followed by the joint statistic for all the parameters in the model (also see §18.4.9), based on the approach in Hansen (1992). Next, the instability statistic is printed for each parameter ( β₁,...,β_k,σ²).

Large values reveal non-constancy (marked by ^* or ^**), and indicate a fragile model. Note that this measures within-sample parameter constancy, and is computed if numerically feasible (it may fail owing to dummy variables), so no observations need be reserved. The indicated significance is only valid in the absence of non-stationary regressors.

18.4.10 Diagnostic tests for NLS

The LM tests for autocorrelation, heteroscedasticity and functional form require an auxiliary regression involving the original regressors x_it. NLS uses ∂f(x_t,θ)/∂θ_i (evaluated at θ̂) instead. The auxiliary regression for the autocorrelation test is:

18.5 Linear restrictions test

Writing the model in matrix form as y=Xβ+u, the null hypothesis of p linear restrictions can be expressed as H₀ : Rβ=r, with R a (p×k) matrix and r a p×1 vector. This test is well explained in most econometrics textbooks, and uses the unrestricted estimates (that is, it is a Wald test).

The subset form of the linear restrictions tests is: H₀: β_i=...=β_j=0: any choice of coefficients can be made, so a wide range of specification hypothesis can be tested.

18.6 General restrictions

Writing θ̂ =β̂, with corresponding variance-covariance matrix V[ θ̂ ] , we can test for (non-) linear restrictions of the form:

The null hypothesis H₀:f(θ)=0 will be tested against H₁:f(θ)≠0 through a Wald test:

where J is the Jacobian matrix of the transformation: J=∂f(θ)/∂θ'. PcGive computes Ĵ by numerical differentiation. The statistic w has a χ²(s) distribution, where s is the number of restrictions (that is, equations in f(^.)). The null hypothesis is rejected if we observe a significant test statistic.

18.7 Test for omitted variables (OLS)

Lag polynomials of any variable in the database can be tested for omission. Variables that would change the sample or are already in the model are automatically deleted. The model itself remains unchanged. If the model is written in matrix form as y=Xβ+Zγ+u, then H₀: γ=0 is being tested. The test exploits the fact that on H₀:

Since ( X'X) ^-1 is precalculated, the F-statistic is easily computed by partitioned inversion. Computations for IVE are more involved.

18.8 Progress: the sequential reduction sequence

Finally, PcGive has specific procedures programmed to operate when a general-to-specific mode is adopted.

In PcGive, when a model is specified and estimated by least squares or instrumental variables, then the general dynamic analysis is offered: see §18.3.

However, while the tests offered are a comprehensive set of Wald statistics on variables, lags and long-run outcomes, a reduction sequence can involve many linear transformations (differencing, creating differentials etc.) as well as eliminations. Consequently, as the reduction proceeds, PcGive monitors its progress, which can be reviewed at the progress menu. The main statistics reported comprise:

18.9 Encompassing and `non-nested' hypotheses tests

Once appropriate data representations have been selected, it is of interest to see whether the chosen model can explain (that is, account for) results reported by other investigators. Often attention has focused on the ability of chosen models to explain each other's residual variances (variance encompassing), and PcGive provides the facility for doing so using test statistics based on Cox (1961) as suggested by Pesaran (1974). Full details of those computed by PcGive for OLS and IVE are provided in Ericsson (1983). Note that a badly-fitting model should be rejected against well-fitting models on such tests, and that care is required in interpreting any outcome in which a well-fitting model (which satisfies all of the other criteria) is rejected against a badly-fitting, or silly, model (see Mizon, 1984, Mizon and Richard, 1986, and Hendry and Richard, 1989). The Sargan test is for the restricted reduced form parsimoniously encompassing the unrestricted reduced form, which is implicitly defined by projecting y_t on all of the non-modelled variables. The F-test is for each model parsimoniously encompassing their union. This is the only one of these tests which is invariant to the choice of common regressors in the two models.

Thus, the F-test yields the same numerical outcome for the first model parsimoniously encompassing either the union of the two models under consideration, or the orthogonal complement to the first model relative to the union. In PcGive, tests of both models encompassing the other are reported.

References

Ahumada, H. (1985). "An encompassing test of two models of the balance of trade for Argentina" Oxford Bulletin of Economics and Statistics, 47, 51--70.

Alexander, C. (2001). Market Models: A Guide to Financial Data Analysis. Chichester: John Wiley and Sons.

Amemiya, T. (1981). "Qualitative response models: A survey" Journal of Economic Literature, 19, 1483--1536.

Anderson, T. W. (1971). The Statistical Analysis of Time Series. New York: John Wiley & Sons.

Andrews, D. W. K. (1991). "Heteroskedasticity and autocorrelation consistent covariance matrix estimation" Econometrica, 59, 817--858.

Baba, Y., Hendry, D. F., and Starr, R. M. (1992). "The demand for M1 in the U.S.A., 1960--1988" Review of Economic Studies, 59, 25--61.

Banerjee, A., Dolado, J. J., Galbraith, J. W., and Hendry, D. F. (1993). Co-integration, Error Correction and the Econometric Analysis of Non-Stationary Data. Oxford: Oxford University Press.

Banerjee, A., Dolado, J. J., Hendry, D. F., and Smith, G. W. (1986). "Exploring equilibrium relationships in econometrics through static models: Some Monte Carlo evidence" Oxford Bulletin of Economics and Statistics, 48, 253--277.

Banerjee, A., Dolado, J. J., and Mestre, R. (1998). "Error-correction mechanism tests for cointegration in a single equation framework" Journal of Time Series Analysis, 19, 267--283.

Banerjee, A., and Hendry, D. F.(eds.)(1992a). Testing Integration and Cointegration. Oxford Bulletin of Economics and Statistics: 54.

Banerjee, A., and Hendry, D. F. (1992b). "Testing integration and cointegration: An overview" Oxford Bulletin of Economics and Statistics, 54, 225--255.

Bårdsen, G. (1989). "The estimation of long run coefficients from error correction models" Oxford Bulletin of Economics and Statistics, 50.

Bentzel, R., and Hansen, B. (1955). "On recursiveness and interdependency in economic models" Review of Economic Studies, 22, 153--168.

Bollerslev, T., Chou, R. S., and Kroner, K. F. (1992). "ARCH modelling in finance -- A review of the theory and empirical evidence" Journal of Econometrics, 52, 5--59.

Bontemps, C., and Mizon, G. E. (2003). "Congruence and encompassing" In Stigum, B. P.(ed.), Econometrics and the Philosophy of Economics, pp. 354--378. Princeton: Princeton University Press.

Bowman, K. O., and Shenton, L. R. (1975). "Omnibus test contours for departures from normality based on √b₁ and b₂" Biometrika, 62, 243--250.

Box, G. E. P., and Jenkins, G. M. (1976). Time Series Analysis, Forecasting and Control. San Francisco: Holden-Day. First published, 1970.

Box, G. E. P., and Pierce, D. A. (1970). "Distribution of residual autocorrelations in autoregressive-integrated moving average time series models" Journal of the American Statistical Association, 65, 1509--1526.

Breusch, T. S., and Pagan, A. R. (1980). "The Lagrange multiplier test and its applications to model specification in econometrics" Review of Economic Studies, 47, 239--253.

Brown, R. L., Durbin, J., and Evans, J. M. (1975). "Techniques for testing the constancy of regression relationships over time (with discussion)" Journal of the Royal Statistical Society B, 37, 149--192.

Campos, J., Ericsson, N. R., and Hendry, D. F. (1996). "Cointegration tests in the presence of structural breaks" Journal of Econometrics, 70, 187--220.

Chambers, E. A., and Cox, D. R. (1967). "Discrimination between alternative binary response models" Biometrika, 54, 573--578.

Chow, G. C. (1960). "Tests of equality between sets of coefficients in two linear regressions" Econometrica, 28, 591--605.

Clements, M. P., and Hendry, D. F. (1998). Forecasting Economic Time Series. Cambridge: Cambridge University Press.

Clements, M. P., and Hendry, D. F. (1999). Forecasting Non-stationary Economic Time Series. Cambridge, Mass.: MIT Press.

Cochrane, D., and Orcutt, G. H. (1949). "Application of least squares regression to relationships containing auto-correlated error terms" Journal of the American Statistical Association, 44, 32--61.

Cox, D. R. (1961). "Tests of separate families of hypotheses" In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, pp. 105--123 Berkeley: University of California Press.

Cramer, J. S. (1986). Econometric Applications of Maximum Likelihood Methods. Cambridge: Cambridge University Press.

D'Agostino, R. B. (1970). "Transformation to normality of the null distribution of g₁" Biometrika, 57, 679--681.

Davidson, J. E. H., Hendry, D. F., Srba, F., and Yeo, J. S. (1978). "Econometric modelling of the aggregate time-series relationship between consumers' expenditure and income in the United Kingdom" Economic Journal, 88, 661--692. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000; and in Campos, J., Ericsson, N.R. and Hendry, D.F. (eds.), General to Specific Modelling. Edward Elgar, 2005.

Davidson, R., and MacKinnon, J. G. (1993). Estimation and Inference in Econometrics. New York: Oxford University Press.

Dickey, D. A., and Fuller, W. A. (1981). "Likelihood ratio statistics for autoregressive time series with a unit root" Econometrica, 49, 1057--1072.

Doan, T., Litterman, R., and Sims, C. A. (1984). "Forecasting and conditional projection using realistic prior distributions" Econometric Reviews, 3, 1--100.

Doornik, J. A. (2007a). "Autometrics" Mimeo, Department of Economics, University of Oxford.

Doornik, J. A. (2007b). Object-Oriented Matrix Programming using Ox 6th ed. London: Timberlake Consultants Press.

Doornik, J. A. (2008). "Encompassing and automatic model selection" Oxford Bulletin of Economics and Statistics, 70, 915--925.

Doornik, J. A. (2009). "Autometrics" In Castle, J. L., and Shephard, N.(eds.), The Methodology and Practice of Econometrics: Festschrift in Honour of David F. Hendry. Forthcoming, Oxford: Oxford University Press.

Doornik, J. A., and Hansen, H. (1994). "A practical test for univariate and multivariate normality" Discussion paper, Nuffield College.

Doornik, J. A., and Hendry, D. F. (1992). PCGIVE 7: An Interactive Econometric Modelling System. Oxford: Institute of Economics and Statistics, University of Oxford.

Doornik, J. A., and Hendry, D. F. (1994). PcGive 8: An Interactive Econometric Modelling System. London: International Thomson Publishing, and Belmont, CA: Duxbury Press.

Doornik, J. A., and Hendry, D. F. (2006). Interactive Monte Carlo Experimentation in Econometrics Using PcNaive 2nd ed. London: Timberlake Consultants Press.

Doornik, J. A., and Hendry, D. F. (2009a). Econometric Modelling using PcGive: Volume III 3rd ed. London: Timberlake Consultants Press.

Doornik, J. A., and Hendry, D. F. (2009b). Modelling Dynamic Systems using PcGive: Volume II 4th ed. London: Timberlake Consultants Press.

Doornik, J. A., and Hendry, D. F. (2009c). OxMetrics: An Interface to Empirical Modelling 6th ed. London: Timberlake Consultants Press.

Doornik, J. A., and Ooms, M. (2006). Introduction to Ox 2nd ed. London: Timberlake Consultants Press.

Durbin, J. (1970). "Testing for serial correlation in least squares regression when some of the regressors are lagged dependent variables" Econometrica, 38, 410--421.

Eicker, F. (1967). "Limit theorems for regressions with unequal and dependent errors" In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, pp. 59--82 Berkeley: University of California.

Eisner, R., and Strotz, R. H. (1963). Determinants of Business Investment. Englewood Cliffs, N.J.: Prentice-Hall.

Emerson, R. A., and Hendry, D. F. (1996). "An evaluation of forecasting using leading indicators" Journal of Forecasting, 15, 271--291. Reprinted in T.C. Mills (ed.), Economic Forecasting. Edward Elgar, 1999.

Engle, R. F. (1982). "Autoregressive conditional heteroscedasticity, with estimates of the variance of United Kingdom inflation" Econometrica, 50, 987--1007.

Engle, R. F. (1984). "Wald, likelihood ratio, and Lagrange multiplier tests in econometrics" in Griliches, and Intriligator 1984, Ch. 13.

Engle, R. F., and Granger, C. W. J. (1987). "Cointegration and error correction: Representation, estimation and testing" Econometrica, 55, 251--276.

Engle, R. F., Hendry, D. F., and Richard, J.-F. (1983). "Exogeneity" Econometrica, 51, 277--304. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000; in Ericsson, N. R. and Irons, J. S. (eds.) Testing Exogeneity, Oxford: Oxford University Press, 1994; and in Campos, J., Ericsson, N.R. and Hendry, D.F. (eds.), General to Specific Modelling. Edward Elgar, 2005.

Engle, R. F., Hendry, D. F., and Trumbull, D. (1985). "Small sample properties of ARCH estimators and tests" Canadian Journal of Economics, 43, 66--93.

Engle, R. F., and White, H.(eds.)(1999). Cointegration, Causality and Forecasting. Oxford: Oxford University Press.

Engler, E., and Nielsen, B. (2009). "The empirical process of autoregressive residuals" Econometrics Journal, forthcoming.

Ericsson, N. R. (1983). "Asymptotic properties of instrumental variables statistics for testing non-nested hypotheses" Review of Economic Studies, 50, 287--303.

Ericsson, N. R. (1992). "Cointegration, exogeneity and policy analysis: An overview" Journal of Policy Modeling, 14, 251--280.

Ericsson, N. R., and Hendry, D. F. (1999). "Encompassing and rational expectations: How sequential corroboration can imply refutation" Empirical Economics, 24, 1--21.

Ericsson, N. R., and Irons, J. S. (1995). "The Lucas critique in practice: Theory without measurement" In Hoover, K. D.(ed.), Macroeconometrics: Developments, Tensions and Prospects, pp. 263--312. Dordrecht: Kluwer Academic Press.

Ericsson, N. R., and MacKinnon, J. G. (2002). "Distributions of error correction tests for cointegration" Econometrics Journal, 5, 285--318.

Escribano, A. (1985). "Non-linear error correction: The case of money demand in the UK (1878--1970)" Mimeo, University of California at San Diego.

Favero, C., and Hendry, D. F. (1992). "Testing the Lucas critique: A review" Econometric Reviews, 11, 265--306.

Finney, D. J. (1947). "The estimation from individual records of the relationship between dose and quantal response" Biometrika, 34, 320--334.

Fletcher, R. (1987). Practical Methods of Optimization, 2nd ed. New York: John Wiley & Sons.

Friedman, M., and Schwartz, A. J. (1982). Monetary Trends in the United States and the United Kingdom: Their Relation to Income, Prices, and Interest Rates, 1867--1975. Chicago: University of Chicago Press.

Frisch, R. (1934). Statistical Confluence Analysis by means of Complete Regression Systems. Oslo: University Institute of Economics.

Frisch, R. (1938). "Statistical versus theoretical relations in economic macrodynamics" Mimeograph dated 17 July 1938, League of Nations Memorandum. Reproduced by University of Oslo in 1948 with Tinbergen's comments. Contained in Memorandum `Autonomy of Economic Relations', 6 November 1948, Oslo, Universitets Økonomiske Institutt. Reprinted in Hendry D. F. and Morgan M. S. (1995), The Foundations of Econometric Analysis. Cambridge: Cambridge University Press.

Frisch, R., and Waugh, F. V. (1933). "Partial time regression as compared with individual trends" Econometrica, 1, 221--223.

Gilbert, C. L. (1986). "Professor Hendry's econometric methodology" Oxford Bulletin of Economics and Statistics, 48, 283--307. Reprinted in Granger, C. W. J. (ed.) (1990), Modelling Economic Series. Oxford: Clarendon Press.

Gilbert, C. L. (1989). "LSE and the British approach to time-series econometrics" Oxford Review of Economic Policy, 41, 108--128.

Godfrey, L. G. (1978). "Testing for higher order serial correlation in regression equations when the regressors include lagged dependent variables" Econometrica, 46, 1303--1313.

Godfrey, L. G. (1988). Misspecification Tests in Econometrics. Cambridge: Cambridge University Press.

Godfrey, L. G., and Orme, C. D. (1994). "The sensitivity of some general checks to omitted variables in the linear model" International Economic Review, 35, 489--506.

Golub, G. H., and Van Loan, C. F. (1989). Matrix Computations. Baltimore: The Johns Hopkins University Press.

Granger, C. W. J. (1969). "Investigating causal relations by econometric models and cross-spectral methods" Econometrica, 37, 424--438.

Granger, C. W. J. (1986). "Developments in the study of cointegrated economic variables" Oxford Bulletin of Economics and Statistics, 48, 213--228.

Granger, C. W. J., and Newbold, P. (1974). "Spurious regressions in econometrics" Journal of Econometrics, 2, 111--120.

Granger, C. W. J., and Newbold, P. (1977). "The time series approach to econometric model building" In Sims, C. A.(ed.), New Methods in Business Cycle Research, pp. 7--21. Minneapolis: Federal Reserve Bank of Minneapolis.

Granger, C. W. J., and Newbold, P. (1986). Forecasting Economic Time Series, 2nd ed. New York: Academic Press.

Gregory, A. W., and Veale, M. R. (1985). "Formulating Wald tests of non-linear restrictions" Econometrica, 53, 1465--1468.

Griliches, Z., and Intriligator, M. D.(eds.)(1984). Handbook of Econometrics, Vol. 2. Amsterdam: North-Holland.

Hansen, B. E. (1992). "Testing for parameter instability in linear models" Journal of Policy Modeling, 14, 517--533.

Harvey, A. C. (1981). The Econometric Analysis of Time Series. Deddington: Philip Allan.

Harvey, A. C. (1990). The Econometric Analysis of Time Series, 2nd ed. Hemel Hempstead: Philip Allan.

Harvey, A. C. (1993). Time Series Models, 2nd ed. Hemel Hempstead: Harvester Wheatsheaf.

Harvey, A. C., and Collier, P. (1977). "Testing for functional misspecification in regression analysis" Journal of Econometrics, 6, 103--119.

Harvey, A. C., and Shephard, N. G. (1992). "Structural time series models" In Maddala, G. S., Rao, C. R., and Vinod, H. D.(eds.), Handbook of Statistics, Vol. 11. Amsterdam: North-Holland.

Hendry, D. F. (1976). "The structure of simultaneous equations estimators" Journal of Econometrics, 4, 51--88. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000.

Hendry, D. F. (1979). "Predictive failure and econometric modelling in macro-economics: The transactions demand for money" In Ormerod, P.(ed.), Economic Modelling, pp. 217--242. London: Heinemann. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000; and in Campos, J., Ericsson, N.R. and Hendry, D.F. (eds.), General to Specific Modelling. Edward Elgar, 2005.

Hendry, D. F. (1980). "Econometrics: Alchemy or science?" Economica, 47, 387--406. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000.

Hendry, D. F.(ed.)(1986a). Econometric Modelling with Cointegrated Variables. Oxford Bulletin of Economics and Statistics: 48.

Hendry, D. F. (1986b). "Econometric modelling with cointegrated variables: An overview" Oxford Bulletin of Economics and Statistics, 48, 201--212. Reprinted in R.F. Engle and C.W.J. Granger (eds), Long-Run Economic Relationships, Oxford: Oxford University Press, 1991, 51--63.

Hendry, D. F. (1986c). "Using PC-GIVE in econometrics teaching" Oxford Bulletin of Economics and Statistics, 48, 87--98.

Hendry, D. F. (1987). "Econometric methodology: A personal perspective" In Bewley, T. F.(ed.), Advances in Econometrics, pp. 29--48. Cambridge: Cambridge University Press. Reprinted in Campos, J., Ericsson, N.R. and Hendry, D.F. (eds.), General to Specific Modelling. Edward Elgar, 2005.

Hendry, D. F. (1988). "The encompassing implications of feedback versus feedforward mechanisms in econometrics" Oxford Economic Papers, 40, 132--149. Reprinted in Ericsson, N. R. and Irons, J. S. (eds.) Testing Exogeneity, Oxford: Oxford University Press, 1994; and in Campos, J., Ericsson, N.R. and Hendry, D.F. (eds.), General to Specific Modelling. Edward Elgar, 2005.

Hendry, D. F. (1989). "Comment on intertemporal consumer behaviour under structural changes in income" Econometric Reviews, 8, 111--121.

Hendry, D. F. (1993). Econometrics: Alchemy or Science? Oxford: Blackwell Publishers.

Hendry, D. F. (1995b). "On the interactions of unit roots and exogeneity" Econometric Reviews, 14, 383--419.

Hendry, D. F. (1995c). "A theory of co-breaking" Mimeo, Nuffield College, University of Oxford.

Hendry, D. F. (1996). "On the constancy of time-series econometric equations" Economic and Social Review, 27, 401--422.

Hendry, D. F. (1997). "On congruent econometric relations: A comment" Carnegie--Rochester Conference Series on Public Policy, 47, 163--190.

Hendry, D. F. (2000a). Econometrics: Alchemy or Science? Oxford: Oxford University Press. New Edition.

Hendry, D. F. (2000b). "Epilogue: The success of general-to-specific model selection" in Econometrics: Alchemy or Science? 2000a, pp. 467--490. New Edition.

Hendry, D. F., and Anderson, G. J. (1977). "Testing dynamic specification in small simultaneous systems: An application to a model of building society behaviour in the United Kingdom" In Intriligator, M. D.(ed.), Frontiers in Quantitative Economics, Vol. 3, pp. 361--383. Amsterdam: North Holland Publishing Company. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000.

Hendry, D. F., and Doornik, J. A. (1994). "Modelling linear dynamic econometric systems" Scottish Journal of Political Economy, 41, 1--33.

Hendry, D. F., and Doornik, J. A. (1997). "The implications for econometric modelling of forecast failure" Scottish Journal of Political Economy, 44, 437--461. Special Issue.

Hendry, D. F., and Ericsson, N. R. (1991). "Modeling the demand for narrow money in the United Kingdom and the United States" European Economic Review, 35, 833--886.

Hendry, D. F., Johansen, S., and Santos, C. (2004). "Selecting a regression saturated by indicators" Unpublished paper, Economics Department, University of Oxford.

Hendry, D. F., and Juselius, K. (2000). "Explaining cointegration analysis: Part I" Energy Journal, 21, 1--42.

Hendry, D. F., and Krolzig, H.-M. (1999). "Improving on `Data mining reconsidered' by K.D. Hoover and S.J. Perez" Econometrics Journal, 2, 202--219. Reprinted in J. Campos, N.R. Ericsson and D.F. Hendry (eds.), General to Specific Modelling. Edward Elgar, 2005.

Hendry, D. F., and Krolzig, H.-M. (2001). Automatic Econometric Model Selection. London: Timberlake Consultants Press.

Hendry, D. F., and Krolzig, H.-M. (2005). "The properties of automatic Gets modelling" Economic Journal, 115, C32--C61.

Hendry, D. F., and Massmann, M. (2007). "Co-breaking: Recent advances and a synopsis of the literature" Journal of Business and Economic Statistics, 25, 33--51.

Hendry, D. F., and Mizon, G. E. (1978). "Serial correlation as a convenient simplification, not a nuisance: A comment on a study of the demand for money by the Bank of England" Economic Journal, 88, 549--563. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000; and in Campos, J., Ericsson, N.R. and Hendry, D.F. (eds.), General to Specific Modelling. Edward Elgar, 2005.

Hendry, D. F., and Mizon, G. E. (1993). "Evaluating dynamic econometric models by encompassing the VAR" In Phillips, P. C. B.(ed.), Models, Methods and Applications of Econometrics, pp. 272--300. Oxford: Basil Blackwell. Reprinted in Campos, J., Ericsson, N.R. and Hendry, D.F. (eds.), General to Specific Modelling. Edward Elgar, 2005.

Hendry, D. F., and Mizon, G. E. (1999). "The pervasiveness of Granger causality in econometrics" in Engle, and White 1999, pp. 102--134.

Hendry, D. F., and Morgan, M. S. (1989). "A re-analysis of confluence analysis" Oxford Economic Papers, 41, 35--52.

Hendry, D. F., and Morgan, M. S. (1995). The Foundations of Econometric Analysis. Cambridge: Cambridge University Press.

Hendry, D. F., and Neale, A. J. (1987). "Monte Carlo experimentation using PC-NAIVE" In Fomby, T., and Rhodes, G. F.(eds.), Advances in Econometrics, Vol. 6, pp. 91--125. Greenwich, Connecticut: Jai Press Inc.

Hendry, D. F., and Neale, A. J. (1988). "Interpreting long-run equilibrium solutions in conventional macro models: A comment" Economic Journal, 98, 808--817.

Hendry, D. F., and Neale, A. J. (1991). "A Monte Carlo study of the effects of structural breaks on tests for unit roots" In Hackl, P., and Westlund, A. H.(eds.), Economic Structural Change, Analysis and Forecasting, pp. 95--119. Berlin: Springer-Verlag.

Hendry, D. F., Neale, A. J., and Ericsson, N. R. (1991). PC-NAIVE, An Interactive Program for Monte Carlo Experimentation in Econometrics. Version 6.0. Oxford: Institute of Economics and Statistics, University of Oxford.

Hendry, D. F., Neale, A. J., and Srba, F. (1988). "Econometric analysis of small linear systems using Pc-Fiml" Journal of Econometrics, 38, 203--226.

Hendry, D. F., and Nielsen, B. (2007). Econometric Modeling: A Likelihood Approach. Princeton: Princeton University Press.

Hendry, D. F., Pagan, A. R., and Sargan, J. D. (1984). "Dynamic specification" in Griliches, and Intriligator 1984, pp. 1023--1100. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000; and in Campos, J., Ericsson, N.R. and Hendry, D.F. (eds.), General to Specific Modelling. Edward Elgar, 2005.

Hendry, D. F., and Richard, J.-F. (1982). "On the formulation of empirical models in dynamic econometrics" Journal of Econometrics, 20, 3--33. Reprinted in Granger, C. W. J. (ed.) (1990), Modelling Economic Series. Oxford: Clarendon Press and in Hendry D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers 1993, and Oxford University Press, 2000; and in Campos, J., Ericsson, N.R. and Hendry, D.F. (eds.), General to Specific Modelling. Edward Elgar, 2005.

Hendry, D. F., and Richard, J.-F. (1983). "The econometric analysis of economic time series (with discussion)" International Statistical Review, 51, 111--163. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000.

Hendry, D. F., and Richard, J.-F. (1989). "Recent developments in the theory of encompassing" In Cornet, B., and Tulkens, H.(eds.), Contributions to Operations Research and Economics. The XXth Anniversary of CORE, pp. 393--440. Cambridge, MA: MIT Press. Reprinted in Campos, J., Ericsson, N.R. and Hendry, D.F. (eds.), General to Specific Modelling. Edward Elgar, 2005.

Hendry, D. F., and Srba, F. (1980). "AUTOREG: A computer program library for dynamic econometric models with autoregressive errors" Journal of Econometrics, 12, 85--102. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000.

Hendry, D. F., and von Ungern-Sternberg, T. (1981). "Liquidity and inflation effects on consumers' expenditure" In Deaton, A. S.(ed.), Essays in the Theory and Measurement of Consumers' Behaviour, pp. 237--261. Cambridge: Cambridge University Press. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000.

Hendry, D. F., and Wallis, K. F.(eds.)(1984). Econometrics and Quantitative Economics. Oxford: Basil Blackwell.

Hooker, R. H. (1901). "Correlation of the marriage rate with trade" Journal of the Royal Statistical Society, 64, 485--492. Reprinted in Hendry, D. F. and Morgan, M. S. (1995), The Foundations of Econometric Analysis. Cambridge: Cambridge University Press.

Hoover, K. D., and Perez, S. J. (1999). "Data mining reconsidered: Encompassing and the general-to-specific approach to specification search" Econometrics Journal, 2, 167--191.

Jarque, C. M., and Bera, A. K. (1987). "A test for normality of observations and regression residuals" International Statistical Review, 55, 163--172.

Johansen, S. (1988). "Statistical analysis of cointegration vectors" Journal of Economic Dynamics and Control, 12, 231--254. Reprinted in R.F. Engle and C.W.J. Granger (eds), Long-Run Economic Relationships, Oxford: Oxford University Press, 1991, 131--52.

Johansen, S. (1995). Likelihood-based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press.

Johansen, S., and Juselius, K. (1990). "Maximum likelihood estimation and inference on cointegration -- With application to the demand for money" Oxford Bulletin of Economics and Statistics, 52, 169--210.

Judd, J., and Scadding, J. (1982). "The search for a stable money demand function: A survey of the post-1973 literature" Journal of Economic Literature, 20, 993--1023.

Judge, G. G., Griffiths, W. E., Hill, R. C., Lütkepohl, H., and Lee, T.-C. (1985). The Theory and Practice of Econometrics, 2nd ed. New York: John Wiley.

Kiviet, J. F. (1986). "On the rigor of some mis-specification tests for modelling dynamic relationships" Review of Economic Studies, 53, 241--261.

Kiviet, J. F. (1987). Testing Linear Econometric Models. Amsterdam: University of Amsterdam.

Kiviet, J. F., and Phillips, G. D. A. (1992). "Exact similar tests for unit roots and cointegration" Oxford Bulletin of Economics and Statistics, 54, 349--367.

Koopmans, T. C.(ed.)(1950). Statistical Inference in Dynamic Economic Models. No. 10 in Cowles Commission Monograph. New York: John Wiley & Sons.

Koopmans, T. C., Rubin, H., and Leipnik, R. B. (1950). "Measuring the equation systems of dynamic economics" in Koopmans 1950, Ch. 2.

Kremers, J. J. M., Ericsson, N. R., and Dolado, J. J. (1992). "The power of cointegration tests" Oxford Bulletin of Economics and Statistics, 54, 325--348.

Kuh, E., Belsley, D. A., and Welsh, R. E. (1980). Regression Diagnostics. Identifying Influential Data and Sources of Collinearity. Wiley Series in Probability and Mathematical Statistics. New York: John Wiley.

Leamer, E. E. (1978). Specification Searches. Ad-Hoc Inference with Non-Experimental Data. New York: John Wiley.

Leamer, E. E. (1983). "Let's take the con out of econometrics" American Economic Review, 73, 31--43. Reprinted in Granger, C. W. J. (ed.) (1990), Modelling Economic Series. Oxford: Clarendon Press.

Ljung, G. M., and Box, G. E. P. (1978). "On a measure of lack of fit in time series models" Biometrika, 65, 297--303.

Lovell, M. C. (1983). "Data mining" Review of Economics and Statistics, 65, 1--12.

Lucas, R. E. (1976). "Econometric policy evaluation: A critique" In Brunner, K., and Meltzer, A.(eds.), The Phillips Curve and Labor Markets, Vol. 1 of Carnegie-Rochester Conferences on Public Policy, pp. 19--46. Amsterdam: North-Holland Publishing Company.

MacKinnon, J. G. (1991). "Critical values for cointegration tests" In Engle, R. F., and Granger, C. W. J.(eds.), Long-Run Economic Relationships, pp. 267--276. Oxford: Oxford University Press.

MacKinnon, J. G., and White, H. (1985). "Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties" Journal of Econometrics, 29, 305--325.

Makridakis, S., Wheelwright, S. C., and Hyndman, R. C. (1998). Forecasting: Methods and Applications 3rd ed. New York: John Wiley and Sons.

Marschak, J. (1953). "Economic measurements for policy and prediction" In Hood, W. C., and Koopmans, T. C.(eds.), Studies in Econometric Method, No. 14 in Cowles Commission Monograph. New York: John Wiley & Sons.

Mizon, G. E. (1977). "Model selection procedures" In Artis, M. J., and Nobay, A. R.(eds.), Studies in Modern Economic Analysis, pp. 97--120. Oxford: Basil Blackwell.

Mizon, G. E. (1984). "The encompassing approach in econometrics" in Hendry, and Wallis 1984, pp. 135--172.

Mizon, G. E. (1995). "A simple message for autocorrelation correctors: Don't" Journal of Econometrics, 69, 267--288.

Mizon, G. E., and Hendry, D. F. (1980). "An empirical application and Monte Carlo analysis of tests of dynamic specification" Review of Economic Studies, 49, 21--45. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000.

Mizon, G. E., and Richard, J.-F. (1986). "The encompassing principle and its application to non-nested hypothesis tests" Econometrica, 54, 657--678.

Nelson, C. R. (1972). "The prediction performance of the FRB-MIT-PENN model of the US economy" American Economic Review, 62, 902--917. Reprinted in T.C. Mills (ed.), Economic Forecasting. Edward Elgar, 1999.

Newey, W. K., and West, K. D. (1987). "A simple positive semi-definite heteroskedasticity and autocorrelation-consistent covariance matrix" Econometrica, 55, 703--708.

Nickell, S. J. (1985). "Error correction, partial adjustment and all that: An expository note" Oxford Bulletin of Economics and Statistics, 47, 119--130.

Nielsen, B. (2006a). "Correlograms for non-stationary autoregressions" Journal of the Royal Statistical Society, B, 68, 707--720.

Nielsen, B. (2006b). "Correlograms for non-stationary autoregressions" Journal of the Royal Statistical Society B, 68, 707--720.

Pagan, A. R. (1984). "Model evaluation by variable addition" in Hendry, and Wallis 1984, pp. 103--135.

Perron, P. (1989). "The Great Crash, the oil price shock and the unit root hypothesis" Econometrica, 57, 1361--1401.

Pesaran, M. H. (1974). "On the general problem of model selection" Review of Economic Studies, 41, 153--171. Reprinted in Campos, J., Ericsson, N.R. and Hendry, D.F. (eds.), General to Specific Modelling. Edward Elgar, 2005.

Phillips, P. C. B. (1986). "Understanding spurious regressions in econometrics" Journal of Econometrics, 33, 311--340.

Phillips, P. C. B. (1987). "Time series regression with a unit root" Econometrica, 55, 277--301.

Phillips, P. C. B. (1991). "Optimal inference in cointegrated systems" Econometrica, 59, 283--306.

Priestley, M. B. (1981). Spectral Analysis and Time Series. London: Academic Press.

Ramsey, J. B. (1969). "Tests for specification errors in classical linear least squares regression analysis" Journal of the Royal Statistical Society B, 31, 350--371.

Richard, J.-F. (1980). "Models with several regimes and changes in exogeneity" Review of Economic Studies, 47, 1--20.

Sargan, J. D. (1958). "The estimation of economic relationships using instrumental variables" Econometrica, 26, 393--415.

Sargan, J. D. (1959). "The estimation of relationships with autocorrelated residuals by the use of instrumental variables" Journal of the Royal Statistical Society B, 21, 91--105. Reprinted as pp. 87--104 in Sargan J. D. (1988), Contributions to Econometrics, Vol. 1, Cambridge: Cambridge University Press.

Sargan, J. D. (1964). "Wages and prices in the United Kingdom: A study in econometric methodology (with discussion)" In Hart, P. E., Mills, G., and Whitaker, J. K.(eds.), Econometric Analysis for National Economic Planning, Vol. 16 of Colston Papers, pp. 25--63. London: Butterworth Co. Reprinted as pp. 275--314 in Hendry D. F. and Wallis K. F. (eds.) (1984). Econometrics and Quantitative Economics. Oxford: Basil Blackwell, and as pp. 124--169 in Sargan J. D. (1988), Contributions to Econometrics, Vol. 1, Cambridge: Cambridge University Press.

Sargan, J. D. (1980a). "The consumer price equation in the post-war British economy. An exercise in equation specification testing" Review of Economic Studies, 47, 113--135.

Sargan, J. D. (1980b). "Some tests of dynamic specification for a single equation" Econometrica, 48, 879--897. Reprinted as pp. 191--212 in Sargan J. D. (1988), Contributions to Econometrics, Vol. 1, Cambridge: Cambridge University Press.

Shenton, L. R., and Bowman, K. O. (1977). "A bivariate model for the distribution of √b₁ and b₂" Journal of the American Statistical Association, 72, 206--211.

Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. London: Chapman and Hall.

Sims, C. A. (1980). "Macroeconomics and reality" Econometrica, 48, 1--48. Reprinted in Granger, C. W. J. (ed.) (1990), Modelling Economic Series. Oxford: Clarendon Press.

Sims, C. A., Stock, J. H., and Watson, M. W. (1990). "Inference in linear time series models with some unit roots" Econometrica, 58, 113--144.

Spanos, A. (1986). Statistical Foundations of Econometric Modelling. Cambridge: Cambridge University Press.

White, H. (1980). "A heteroskedastic-consistent covariance matrix estimator and a direct test for heteroskedasticity" Econometrica, 48, 817--838.

White, H. (1984). Asymptotic Theory for Econometricians. London: Academic Press.

White, H. (1990). "A consistent model selection" In Granger, C. W. J.(ed.), Modelling Economic Series, pp. 369--383. Oxford: Clarendon Press.

Wooldridge, J. M. (1999). "Asymptotic properties of some specification tests in linear models with integrated processes" in Engle, and White 1999, pp. 366--384.

Working, E. J. (1927). "What do statistical demand curves show?" Quarterly Journal of Economics, 41, 212--235.

Yule, G. U. (1926). "Why do we sometimes get nonsense-correlations between time-series? A study in sampling and the nature of time series (with discussion)" Journal of the Royal Statistical Society, 89, 1--64. Reprinted in Hendry, D. F. and Morgan, M. S. (1995), The Foundations of Econometric Analysis. Cambridge: Cambridge University Press.

coefficients	α̂ and μ̂ (if included), β̂, γ̂₁,...,γ̂_n,
standard errors	SE(α̂), SE(μ̂), SE(β̂), SE(γ̂_i),
t-values	t_α , t_μ , t_β , t_{γ_i},
σ̂	as (eq:17.10),
DW	(eq:16.3) applied to û_t,
DW(x)	(eq:16.3) applied to x_t,
ADF(x)	t_β ,
Critical values
RSS	as (eq:17.11).

D-lag	j (the number of lagged differences),
t-adf	the t-value on the lagged level: t_β ,
beta Y_1	the coefficient on the lagged level: β,
σ̂	as (eq:17.10),
t-DY_lag	t-value of the longest lag: t_{γ_j},
t-prob	significance of the longest lag: 1-P( \| τ\| ≤\| t_{γ_j}\| ) ,
AIC	Akaike criterion, see §17.2.12
F-prob	significance level of the F-test on the lags dropped up to that point,

	nominal probabilities of e₂				nominal probabilities of (eq:18.18)
T	20%	10%	5%	1%	20%	10%	5%	1%
50	0.1734	0.0869	0.0450	0.0113	0.0939	0.0547	0.0346	0.0175
100	0.1771	0.0922	0.0484	0.0111	0.1258	0.0637	0.0391	0.0183
150	0.1845	0.0937	0.0495	0.0131	0.1456	0.0703	0.0449	0.0188
250	0.1889	0.0948	0.0498	0.0133	0.1583	0.0788	0.0460	0.0180

OLS-CS	ordinary least squares (cross-section modelling)
IVE-CS	instrumental variables estimation (cross-section modelling)
OLS	ordinary least squares
IVE	instrumental variables estimation
RALS	r^th order autoregressive least squares
NLS	non-linear least squares
ML	maximum likelihood estimation

RLS	recursive OLS
RIVE	recursive IVE
RNLS	recursive NLS
RML	recursive ML

mean	x
standard deviation	σ_x=(m₂)^½
skewness	√b₁
excess kurtosis	b₂-3
minimum
maximum
asymptotic test	e₁
normality testχ²( 2)	e₂ [ P( χ²( 2) ≥e₂) ]

PcGive Volume I: Empirical Econometric Modelling

Part:6 The Statistical Output of PcGive

Chapter 16 Descriptive Statistics in PcGive