PcGets Help on Model Estimation Statistics

Once a model has been specified, a sample period selected, and an estimation method chosen, the equation can be estimated. 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) whereas the H retained observations X_H=( x_T+1...x_T+H) are used for in-sample forecasts. Recursive estimation methods are initialized by a direct estimation over t=1,...,M-1, followed by recursive estimation over t=M,...,T (or the reverse for backward recursion).

This chapter discusses the statistics reported by PcGets following model estimation. The next chapter describes the evaluation tools. Sections marked with * denote information that can be requested.

12.2 Model formulation

We first consider the estimation of linear equations by ordinary least squares (OLS) and instrumental variables (IVs), as these are the main methods offered by PcGets. Chapter 10 considered the econometrics of model selection: here we are primarily concerned with the formulae needed to estimate any given specification by OLS or IVE. Below, matrices and vectors are in bold. For distributional results, we assume the data are either stationary or have been transformed to a non-integrated representation.

A regression is the mean of the conditional distribution of one random variable, denoted y, given the values of k others, denoted x, where x=( x₁...x_k) '. In a multivariate normal distribution, regressions are linear (and homoscedastic), so have the form:

One of the x_i,t will generally be a constant (equal to unity), with the corresponding parameter being the intercept in (eq:12.1). b=( b₁...b_k) 'ÎĀ^k, is the k×1 parameter vector of interest. Usually, a relation like (eq:12.1) is expressed in model form as (for t=1,...,T):

The unknown parameters b and s_e² (the equation standard error) must be estimated from the available sample of data. Section 12.3 describes how.

Equations like (eq:12.2) could also be postulated without the justification that they are the conditional mean of a normal distribution. If, nevertheless, E[ y_t|x_t] =b'x_t then OLS remains consistent. However, when e_t is not independent of x_t, then sufficient instrumental variables, denoted z_t, are required such that E[ e_t|z_t] =0, when rankE[ z_tz_t'] ³n for n endogenous right-hand side regressors. IV estimators are considered in section 12.6 below.

Since linear models play a major role in Gets, §12.3 formulates regression estimation, beginning with the algebra of ordinary least squares in §12.3.1. Some intermediate results on idempotent matrices and distributions of functions of normal variables in §12.3.5 and §12.3.6 respectively lead in §12.3.7 to the distributional results which underpin inference. Tests on subsets of estimates are developed in §12.3.9, and recursive estimation is noted in §12.4.

Section 12.6 then develops the algebra of IVE estimation and inference, and concludes by noting recursive estimation.

12.3 OLS estimation

The linear model under analysis is given by (eq:12.2). Grouping the observations, so that y'=( y₁...y_T) and X'=( x₁...x_T) , which is a T×k matrix with rank( X) =k and e^'=( e₁...e_T) , then:

where E[ X'e] =0. Although conditioning on X is too strong to be justifiable in economics, and essentially entails an experimental setting, most of the results recorded below hold in large samples under much weaker assumptions (see e.g., Hendry, 1995a), and can be extended to integrated processes (see e.g., Sims, Stock and Watson, 1990, and Banerjee, Dolado, Galbraith and Hendry, 1993). The assumptions about e are almost equally strong, but less objectionable in practice given the ability of investigators to formulate congruent models: see the discussion in Chapters 9 and 10.

12.3.1 The algebra of ordinary least squares

The algebra of OLS estimation does not depend on the statistical status of X, and the formulae below -- which are programmed in computer packages -- are often applied even when conditioning is invalid. However, statistical results do depend on the status of X, and the assumptions about e.

Letting e^{^}=y-Xb^{^} then X'(y-Xb^{^})=X'e^{^}=0. To determine the properties of b^{^} as an estimator of b, when (eq:12.2) holds, substitute for y from (eq:12.4):

Thus, b^{^} is unbiased for b. More generally, when x_t is weakly exogenous for b in (eq:12.2), then b^{^} is consistent for b.

12.3.2 *Correlations

The correlation matrix (with diagonal equal to unity) is reported for both the general and specific models. The second-moment matrix M_XX=T^-1X'X, and the matrix of correlations:

12.3.3 Estimated regression equation

The first column of output records the names of the variables and the second, the estimated regression coefficients b^{^} in (eq:12.5). PcGets uses the QR decomposition with partial pivoting rather than the formula in (eq:12.5) to calculate b^{^}, which analytically gives the same result, but is numerically more stable. The QR decomposition of X is X=QR, where Q is T×T and orthogonal (so Q'Q=I_T), and R is T×k and upper triangular, so X'X=R'R.

The following three columns give further information about each of the magnitudes described below in §12.3.4 and §12.3.10.

12.3.4 Variances

and e^{^}'e^{^}= RSS (an acronym for residual sum of squares). In turn, letting V[ ^.] denote variance, V[b^{^}] can be estimated by:

12.3.4.1 Standard errors of the regression coefficients

where d_ii is the i^th diagonal element of ( X'X) ^-1 and s^{^}_u is the standard error of the regression, defined in §12.3.13 below.

12.3.5 Some intermediate algebra

Let M=I_T-X( X'X) ^-1X', which is a symmetric and idempotent T×T matrix, such that M=M', M=M² and M( I_T-M) =0. Further, M annihilates X since MX=0, so that:

where the last equality follows from pre-multiplying (eq:12.4) by M. Consequently:

Since M is real and symmetric, let M=HLH^' where L is the diagonal matrix of eigenvalues and H is the non-singular matrix of eigenvectors with H'H=I_T. By idempotency:

so L²=L and all the eigenvalues of M are either zero or unity. Thus, rank ( M) =tr( M) =tr( L) so:

12.3.6 Functions of normal variables: c², t and F distributions

Three important functions of normally distributed random variables are distributed as the c², t, and F distributions. First, we define these three distributions, then consider generalizations. Let Z~N[ 0,1] then:

where c²( 1) is the chi-squared distribution with one degree of freedom. For a set of k independent random variables Z_i~IN[ 0,1] :

Thirdly, let h₁( k₁) and h₂( k₂) be two independent chi-squareds of k₁ and k₂ degrees of freedom, then:

where F(k₁,k₂) is the F-distribution with k₁ and k₂ degrees of freedom. Note that t(k)²~F(1,k) by using these three results. All of these distributions have been tabulated (in more modern terms, programmed into computer packages), and occur frequently in empirical research, in the sense that an underlying normal distribution is often assumed.

This result follows from the definition of a c² by noting that any positive definite matrix S can be written as S=HH' where H is a non-singular lower triangular matrix, so that:

Partition z' into the k₁ and k₂ independent components ( z₁':z₂^') , each of which is normal, then for k₁+k₂=k:

12.3.7 Distributional results

Since e~N_T[0,s_u²I], then e'e/s_u²~c²( T) . Because M is singular, we cannot apply the theorems of §12.3.6 on the distributions of functions of normal variables to Me or e'Me. However, collect all the unit eigenvalues of M in the first ( T-k) diagonal elements of L, with the last k diagonal elements being zeros. Let:

where n'=( n₁':n₂') and n₁ and n₂ correspond to the unit and zero roots respectively in L, so that n₁ denotes the first ( T-k) elements of n corresponding to the unit eigenvalues of M. Then n₁~N_T-k[0,s_u²I], and since n~N_T[0,s_u²I], n₁ and n₂ are distributed independently. Hence:

Thus, an idempotent quadratic form in standardized normal variables is distributed as a c² with degrees of freedom equal to the rank of the idempotent matrix. Also:

The properties of s^{^}_e² can be calculated from this last result using the c²-distribution. As h₂~c²( T-k) , then E[ h₂] =T-k and V[ h₂] =2( T-k) .

Next, b^{^} is a linear function of the normally distributed vector e from (eq:12.4)--(eq:12.9:

so h₁=n₂'n₂/s_e². As n₁ and n₂ are distributed independently, h₁ and h₂ are also independent, matching their being c²( k) and c²( T-k) respectively.

Tests of H₀: b=0 (or components thereof) follow from these results using the F-distribution, since y=e when b=0, so that on H₀:

The last expression for h_b is the statistic which is actually computed to test H₀ for the GUM. If b¹0 , then from (eq:12.25) the numerator of h_b becomes a non-central c² with non-centrality parameter b'X'Xb³0, whereas the denominator is unchanged. Thus, the statistic h_b will on average lead to values larger than the F( k,T-k) anticipated under the null.

Finally, from (eq:12.25) each element b^{^}_i of b^{^} is normally distributed with variance given by s_u² times the i^th diagonal element d_ii of ( X'X) ^-1 and is independent of h₂. Thus:

where t( T-k) denotes Student's t-distribution with ( T-k) degrees of freedom and:

12.3.8 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 b_i=0. The null hypothesis is rejected if the probability of getting a t-value at least as large is less than the chosen significance level. This probability is given as:

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, t 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.

12.3.9 Subsets of parameters

Consider estimating a subset of k_b parameters b_b of b where k_a+k_b=k. Partition X=( X_a:X_b) and b'=(b_a':b_b') so that:

Consequently, b^{^}_b can be calculated by first regressing X_b on X_a and saving the residuals (M_aX_b), then regressing y on those residuals. Notice that the regressors need `corrected', but the regressand does not. Thus, procedures for `detrending' (for example), can be justified as equivalent to adding a trend to the model (see Frisch and Waugh, 1933, who first proved this famous result). From (eq:12.36) and (eq:12.37):

Hypothesis tests about b_b follow analogously to the previous section. In particular, from §12.3.7 and (eq:12.40):

When k_b=1, this matches (eq:12.32) under H₀: b_b=0, and when k_b=k, (eq:12.41) reproduces (eq:12.29) under H₀: b=0. A useful case of (eq:12.41) is k_b=( k-1) and X_a=i (a T×1 vector of ones), so all coefficients other than the intercept are tested.

If, instead of estimating (eq:12.35), X_a is omitted from the model in the incorrect belief that b_a=0, the equation to be estimated becomes:

The resulting estimator of b_b, denoted by b^~_b=( X_b'X_b) ^-1X_b'y, confounds the effects of X_a and X_b:

which equals b_b if and only if B_bab_a=0. Moreover, let M_b have the same form as M_a but using X_b then:

Conventionally, b^~_b is interpreted as a biased estimator of b_b with bias given by B_bab_a. The estimated variance matrix in (eq:12.45) may exceed, or be less than, that given by the relevant sub-matrix of (eq:12.12), in the sense that the difference could be positive or negative semi-definite.

12.3.10 *Parameter reliability statistics

If the `sample split analysis' option of the Model Settings dialog is marked, for each parameter ( b₁,...,b_k) in the final model, a reliability statistic is computed, based on its significance in two overlapping sub-samples as well as the whole sample. The final column records these measures for the bs. Low values may indicate a variable that is significant by chance in the whole sample, or one that is retained to offset a potential diagnostic-test problem.

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

12.3.11 Log-likelihood (LogLik)

12.3.12 R²: squared multiple correlation coefficient

assuming a constant is included. Intercepts should usually be included in the initial selection of x_t, and PcGets will decide on its final retention.

12.3.13 Equation standard error ( s^{^}_e)

The equation standard error is the square root of the residual variance, which is defined as:

12.3.14 Residual sum of squares (RSS)

12.3.15 Information criteria

The three statistics reported are the Schwarz criterion (SC), the Hannan--Quinn (HQ) criterion, and the Akaike criterion (AIC). Here:

12.4 Recursive OLS estimation

where b is assumed to be constant, E[ x_te_t] =0"t, E[ e_t²] =s_e², and E[ e_te_s] =0 if t¹s. Let the complete sample period be ( 1,...,T) , and consider the least-squares outcome on a sub-sample up to t-1 (for t>k when there are k regressors in x_t-1):

with X_t-1=( x₁...x_t-1) ' and y_t-1=( y₁...y_t-1) '. When the sample is increased by one observation, then:

Thus, b^{^}_t and RSS _t follow directly, from which equation and parameter standard errors are readily calculated:

A new calculation is the set of innovations, which are the one-step ahead forecast errors:

These are mean zero, independent random variables, but with heteroscedastic variance.

Finally, from the sequence of {RSS_t-1}, sequences of tests (for example, for parameter constancy) can be calculated, based on Chow (1960). Both forwards and backwards recursive estimates are provided, as are moving window estimates (i.e., a fixed sample size).

12.5 Forecasting

Forecasting may be done 1-step ahead or h-steps ahead. The former are ex post (or static): any lagged information required to form forecasts is based on observed values. The latter are ex ante (or dynamic) and will reuse forecasts from previous period(s) if required. Consider the model:

estimated over t=1,...,T conditional on z_t. Assuming a forecast horizon of H=3, we find:

Both types of forecast require data for t=1,...,T to obtain the coefficient estimates. Beyond that, static forecasts require z for t=T+1,...,T+H and y up to T+H-1, whereas dynamic forecasts only need z for t=T+1,...,T+H. In both types of forecast, the parameter estimates are fixed over the forecast period.

To distinguish the two types of forecast, let y^{^}_T+i,h denote the h-step forecast made for period T+i (i³h) and based on parameter estimation up to T. The h-step forecasts use actual values for lagged ys which go further back than h periods. Thus, 1-step forecasts, y^{^}_T+h,1 (h=1,...,H), always use actual values for lagged ys. Dynamic forecasts, y^{^}_T+h,h (h=1,...,H), never use actual values beyond T. Intermediate (h) step forecasts, using the new notation and omitting the zs, are:

12.5.1 Analysis of 1-step forecasts

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, see §12.4. 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 (y^{^}_t), the forecast error (e_t=y_t-y^{^}_t), the standard error of the 1-step forecast (SE[ e_t] =ÖV[ e_t] ^{^}), and a t-value (that is, the standardized forecast error e_t/SE[ e_t] ).

12.5.2 Dynamic forecasting

To derive expressions for dynamic forecasts, we need to take the lag structure of y into account. Consider a simple model with one lag of the dependent variable:

since E[ u_iu_j] =0 for i¹j and E[ u_i²] =s_u². Similar expressions hold when there are m lags of the dependent variable. For appropriate formulae when parameter-estimation variances are included, see Doornik and Hendry (2001a).

The model must characterize the economy as accurately in the forecast period as it did over the estimation sample if the forecast errors are to be from the same distribution as that assumed in-sample. This is a strong requirement, and seems unlikely to be met unless the parameters are constant within sample. Clements and Hendry (1998) give forecast-error variances for the whole sequence of forecasts, and Clements and Hendry (1999a) discuss forecasting with non-stationary series.

12.6 Instrumental variables estimation

The method of instrumental variables was originally developed as a computationally-simple approach for handling endogenous regressors in simultaneous-equations models. Although system procedures now abound (see the estimator generating approach in Hendry, 1976), it remains convenient to have methods that are viable for an individual equation.

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^*. 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. Thus, there are k=n-1+q₁ coefficients of interest b=(b₀':b₁')'. Letting x_t=(y_t^*':w_t')', with X'=( x₁...x_T) , y=(y₁...y_T)', then the model is:

12.6.1 The algebra of IVE

When E[ y_t|x_t] ¹x_t'b so that E[ x_te_t] ¹0 in (eq:12.2), then OLS estimation is inconsistent for b. If there is a n×1 vector z_t such that n³k and:

so E[ z_t'e_t] =0, then z_t can be used as an instrumental variable in estimating b (b is not estimable when there are fewer than k instruments). Collecting all the observations:

When rank ( T^-1Z'X) =k, which is also sufficient to identify b (see White, 1984), and rank( T^-1Z'Z) =n, then setting Z'e=0 (noting E[ Z'e] =0):

which is the generalized instrumental variables estimator. As for OLS, PcGets does not use expression (eq:12.74) directly, but instead uses the QR decomposition to ensure the computation is numerically more stable. If n=k, equation (eq:12.74) simplifies to:

denote the `reduced form' of the regressors as a linear function of the instruments, then, as E[ Z'V] =0:

so rank(P^{^})=k ensures identification (the `quality' of the estimates depends on how `large' P^{^} is, using a small number of additional instruments).

12.6.2 Testing hypotheses on b

A c² test of b=0 (other than the intercept) can be calculated which has a crude correspondence to the earlier F-test for OLS. On H₀: b=0, the reported statistic behaves asymptotically as a c²( k-1) . Let:

would test whether all k coefficients are zero. To keep the intercept separate, we compute:

This amounts to using the formula for b^~ in x_b with y-yi instead of y, where i is a vector of unit elements. An approximate F-test can be formed as in Kiviet (1986).

Tests on subsets follow similarly. Individual `t'-statistics can be based on the ratios of estimated coefficients to their estimated standard errors from (eq:12.77).

12.6.3 Specification c²

This tests for the validity of the choice of the instrumental variables as discussed by Sargan (1964). It is asymptotically distributed as c²(q) when the q over-identifying instruments are independent of the equation error. It is also interpretable as a test of whether the restricted reduced form of the structural model (y_t on x_t combined with x_t on z_t) parsimoniously encompasses the unrestricted reduced form (y_t on z_t directly). Let:

with p^{^}=( Z'Z) ^-1Z'y being the `reduced form' estimates. From (eq:12.2) and (eq:12.76):

12.6.4 Recursive IV estimation

When instrumental variables estimators are used, the recursive formulae are similar to OLS but more cumbersome (see Hendry and Neale, 1987). PcGets simply recalculates the estimates as the sample size is changed. Like OLS, both forwards and backwards recursive estimates are provided, as are moving window.

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OLS	Ordinary Least Squares Estimation
GETS	Testimation
IVE	Instrumental Variables Estimation
GETSIVE	Instrumental Variables Testimation

Model Estimation Statistics

Contents

Chapter 12 Model Estimation Statistics

12.1 Introduction