Types of diagnostic tests with null PDF

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Specification and Diagnostic Tests Empirical research is usually an interactive process. The process begins with a specification of the relationship to be estimated. Selecting a specification usually involves several choices: the variables to be included, the functional form connecting these variables, and if the data are time series, the dynamic structure of the relationship between the variables. Inevitably, there is uncertainty regarding the appropriateness of this initial specification. Once you estimate your equation, EViews provides tools for evaluating the quality of your specification along a number of dimensions. In turn, the results of these tests influence the chosen specification, and the process is repeated. Here we describe the extensive menu of specification test statistics that are available as views or procedures of an equation object. While we attempt to provide you with sufficient statistical background to conduct the tests, practical considerations ensure that many of the descriptions are incomplete. We refer you to standard statistical and econometric references for further details.

Background Each test procedure described below involves the specification of a null hypothesis, which is the hypothesis under test. Output from a test command consists of the sample values of one or more test statistics and their associated probability numbers (p-values). The latter indicate the probability of obtaining a test statistic whose absolute value is greater than or equal to that of the sample statistic if the null hypothesis is true. Thus, low p-values lead to the rejection of the null hypothesis. For example, if a p-value lies between 0.05 and 0.01, the null hypothesis is rejected at the 5 percent but not at the 1 percent level. Bear in mind that there are different assumptions and distributional results associated with each test. For example, some of the test statistics have exact, finite sample distributions (usually t or F -distributions). Others are large sample test statistics with asymptotic distributions. Details vary from one test to another and are given below in the description of each test.

Types of Tests The View button on the equation toolbar gives you a choice among three categories of tests to check the specification of the equation. Additional tests are discussed elsewhere: these include unit root tests, the Granger causality test, tests specific to binary, ordered, censored, and count models, and the Johansen's test for cointegration.

Coefficient Tests These tests evaluate restrictions on the estimated coefficients, including the special case of tests for omitted and redundant variables.

Wald Test — Coefficient Restrictions The Wald test computes the test statistic by estimating the unrestricted regression without imposing the coefficient restrictions specified by the null hypothesis. The Wald statistic measures how close the unrestricted estimates come to satisfying the restrictions under the null hypothesis. If the restrictions are in fact true, then the unrestricted estimates should come close to satisfying the restrictions. We first give a general formula for the computation of the Wald test statistic; users who are familiar with the formula or wish to skip the technical details can skip this subsection. Details Consider a general nonlinear regression model

where

is a k vector of parameters to estimate. Any restrictions on the parameters can be written as ,

where g is a smooth q dimensional vector imposing q restrictions on

. The Wald statistic is then

computed as

where n is the number of observations and b is the unrestricted parameter estimates. V is the estimated variance of b given by

where u is the unrestricted residuals. More formally, under the null hypothesis

, the Wald statistic has an asymptotic

where q is the number of restrictions under

(q) distribution,

.

The Wald test can be applied to equations estimated by least squares, two-stage least squares, nonlinear least squares, binary, ordered, censored, truncated, and count models, and systems estimators. The Wald test is the only test described here that can be applied to equations estimated by system methods. For the special case of a linear regression model

and linear restrictions , where R is a known q k matrix, and r is a q vector, respectively, the Wald statistic reduces to , which is asymptotically distributed as If we further assume that the errors exact, finite sample F-statistic:

(q) under

.

are independent and identically normally distributed, we have an

, where is the vector of residuals from the restricted regression. The F-statistic compares the residual sum of squares computed with and without the restrictions imposed. If the restrictions are valid, there should be little difference in the two residual sum of squares and the F-value should be small. EViews reports both the chi-square and the F-statistics and the associated p-values. How to Perform Wald Coefficient Tests To demonstrate how to perform Wald tests, we consider an example. Suppose a Cobb-Douglas production function has been estimated in the form: , where Q, K and L denote value-added output and the inputs of capital and labor respectively. The hypothesis of constant returns to scale is then tested by the restriction: + =1. Estimation of the Cobb-Douglas production function using annual data from 1947 to 1971 provided the following result: Dependent Variable: LOG(Q) Method: Least Squares Date: 08/11/97 Time: 16:56 Sample: 1947 1971 Included observations: 25

Variable C LOG(L) LOG(K)

Coefficient -2.327939 1.591175 0.239604

Std. Error 0.410601 0.167740 0.105390

t-Statistic -5.669595 9.485970 2.273498

Prob. 0.0000 0.0000 0.0331

The sum of the coefficients on LOG(L) and LOG(K) appears to be in excess of one, but to determine whether the difference is statistically relevant, we will conduct the hypothesis test of constant returns. To carry out a Wald test, choose View/Coefficient Tests/Wald-Coefficient Restrictions… from the equation toolbar. Enter the restrictions into the edit box, with multiple coefficient restrictions separated by commas. The restrictions should be expressed as equations involving the estimated coefficients and constants (you may not include series names). The coefficients should be referred to as C(1), C(2), and so on, unless you have used a different coefficient vector in estimation. To test the hypothesis of constant returns to scale, type the following restriction in the dialog box: c(2) + c(3) = 1 and click OK. EViews reports the following result of the Wald test: Wald Test: Equation: EQ1 Null Hypothesis: F-statistic Chi-square

C(2)+C(3)=1 120.0177 120.0177

Probability Probability

0.000000 0.000000

EViews reports an F-statistic and a Chi-square statistic with associated p-values. The Chi-square statistic is equal to the F-statistic times the number of restrictions under test. In this example, there is only one restriction and so the two test statistics are identical with the p-values of both statistics indicating that we can decisively reject the null hypothesis of constant returns to scale. To test more than one restriction, separate the restrictions by commas. For example, to test the hypothesis that the elasticity of output with respect to labor is 2/3 and with respect to capital is 1/3, type the restrictions as c(2)=2/3, c(3)=1/3 and EViews reports Wald Test: Equation: DEMAND Null Hypothesis: F-statistic Chi-square

C(2)=2/3 C(1)=1/3 385.6769 771.3538

Probability Probability

0.000000 0.000000

As an example of a nonlinear model with a nonlinear restriction, we estimate a production function of the form

and test the constant elasticity of substitution (CES) production function restriction . This is an example of a nonlinear restriction. To estimate the (unrestricted) nonlinear model, you should select Quick/Estimate Equation… and then enter the following specification: log(q) = c(1) + c(2)*log(c(3)*k^c(4)+(1-c(3))*l^c(4)) To test the nonlinear restriction, choose View/Coefficient Tests/Wald-Coefficient Restrictions… from the equation toolbar and type the following restriction in the Wald Test dialog box: c(2)=1/c(4)

The results are presented below: Wald Test: Equation: EQ2 Null Hypothesis: F-statistic Chi-square

C(2)=1/C(4) 0.028507 0.028507

Probability Probability

0.867539 0.865923

Since this is a nonlinear test, we focus on the Chi-square statistic which fails to reject the null hypothesis. It is well-known that nonlinear Wald tests are not invariant to the way that you specify the nonlinear restrictions. In this example, the nonlinear restriction can equivalently be written as or . For example, entering the restriction as c(2)*c(4)=1 yields: Wald Test: Equation: EQ2 Null Hypothesis: F-statistic Chi-square

C(2)*C(4)=1 104.5599 104.5599

Probability Probability

0.000000 0.000000

The test now decisively rejects the null hypothesis. We hasten to add that this is not a situation that is unique to EViews, but is a more general property of the Wald test. Unfortunately, there does not seem to be a general solution to this problem; see Davidson and MacKinnon (1993, Chapter 13) for further discussion and references. An alternative is to use the LR test, which is invariant to reparameterization. To carry out the LR test, you compute both the unrestricted model (as estimated above) and the restricted model (imposing the nonlinear restrictions) and compare the log likelihood values. We will estimate the two specifications using the command form. The restricted model can be estimated by the entering the following command in the command window: equation eq_ces0.ls log(q) = c(1) + c(2)*log(c(3)*k^(1/c(2))+(1-c(3))*l^(1/c(2))) while the unrestricted model is estimated using the command: equation eq_ces1.ls log(q) = c(1) + c(2)*log(c(3)*k^c(4)+(1-c(3))*l^c(4)) Note that we save the two equations in the named equation objects EQ0 and EQ1. The LR test statistic can then be computed as scalar lr=-2*(eq_ces0.@logl-eq_ces1.@logl) To see the value of LR, double click on LR; the value will be displayed in the status line at the bottom of the EViews window. You can also compute the p-value of LR by the command scalar lr_pval=1-@cchisq(lr,1)

Omitted Variables This test enables you to add a set of variables to an existing equation and to ask whether the set makes a significant contribution to explaining the variation in the dependent variable. The null hypothesis is that the additional set of regressors are not jointly significant. The output from the test is an F-statistic and a likelihood ratio (LR) statistic with associated p-values, together with the estimation results of the unrestricted model under the alternative. The F-statistic is based on the difference between the residual sums of squares of the restricted and unrestricted

regressions. The LR statistic is computed as

where

and

are the maximized values of the (Gaussian) log likelihood function of the unrestricted

and restricted regressions, respectively. Under , the LR statistic has an asymptotic distribution with degrees of freedom equal to the number of restrictions, i.e. the number of added variables. Bear in mind that: The omitted variables test requires that the same number of observations exist in the original and test equations. If any of the series to be added contain missing observations over the sample of the original equation (which will often be the case when you add lagged variables), the test statistics cannot be constructed. The omitted variables test can be applied to equations estimated with linear LS, TSLS, ARCH (mean equation only), binary, ordered, censored, truncated, and count models. The test is available only if you specify the equation by listing the regressors, not by a formula. To perform an LR test in these settings, you can estimate a separate equation for the unrestricted and restricted models over a common sample, and evaluate the LR statistic and p-value using scalars and the @cchisq function, as described above. How to Perform an Omitted Variables Test To test for omitted variables, select View/Coefficient Tests/Omitted Variables-Likelihood Ratio… In the dialog that opens, list the names of the test variables, each separated by at least one space. Suppose, for example, that the initial regression is ls log(q) c log(l) log(k) If you enter the list log(m) log(e) in the dialog, then EViews reports the results of the unrestricted regression containing the two additional explanatory variables, and displays statistics testing the hypothesis that the coefficients on the new variables are jointly zero. The top part of the output depicts the test results: Omitted Variables: LOG(M) LOG(E) F-statistic Log likelihood ratio

4.267478 Probability 8.884940 Probability

0.028611 0.011767

The F-statistic has an exact finite sample F-distribution under if the errors are independent and identically distributed normal random variables. The numerator degrees of freedom is the number of additional regressors and the denominator degrees of freedom is the number of observations less the total number of regressors. The log likelihood ratio statistic is the LR test statistic and is asymptotically distributed as a

with degrees of freedom equal to the number of added regressors.

In our example, the tests reject the null hypothesis that the two series do not belong to the equation at a 5% significance level, but cannot reject the hypothesis at a 1% significance level.

Redundant Variables The redundant variables test allows you to test for the statistical significance of a subset of your included variables. More formally, the test is for whether a subset of variables in an equation all have zero coefficients and might thus be deleted from the equation. The redundant variables test can be applied to equations estimated by linear LS, TSLS, ARCH (mean equation only), binary, ordered, censored, truncated, and count methods. The test is available only if you specify the equation by listing the regressors, not by a formula. How to Perform a Redundant Variables Test To test for redundant variables, select View/Coefficient Tests/Redundant Variables-Likelihood

Ratio… In the dialog that appears, list the names of each of the test variables, separated by at least one space. Suppose, for example, that the initial regression is ls log(q) c log(l) log(k) log(m) log(e) If you type the list log(m) log(e) in the dialog, then EViews reports the results of the restricted regression dropping the two regressors, followed by the statistics associated with the test of the hypothesis that the coefficients on the two variables are jointly zero. The test statistics are the F-statistic and the Log likelihood ratio. The F-statistic has an exact finite sample F-distribution under if the errors are independent and identically distributed normal random variables. The numerator degrees of freedom are given by the number of coefficient restrictions in the null hypothesis. The denominator degrees of freedom are given by the total regression degrees of freedom. The LR test is an asymptotic test, distributed as a with degrees of freedom equal to the number of excluded variables under . In this case, there are two degrees of freedom.

Residual Tests EViews provides tests for serial correlation, normality, heteroskedasticity, and autoregressive conditional heteroskedasticity in the residuals from your estimated equation. Not all of these tests are available for every specification.

Correlograms and Q-statistics This view displays the autocorrelations and partial autocorrelations of the equation residuals up to the specified number of lags. Further details on these statistics and the Ljung-Box Q-statistics that are also computed are provided in Series Views, Correlogram. This view is available for the residuals from least squares, two-stage least squares, nonlinear least squares and binary, ordered, censored, and count models. In calculating the probability values for the Q-statistics, the degrees of freedom are adjusted to account for estimated ARMA terms. To display the correlograms and Q-statistics, push View/Residual Tests/Correlogram-Q-statistics on the equation toolbar. In the Lag Specification dialog box, specify the number of lags you wish to use in computing the correlogram.

Correlograms of Squared Residuals This view displays the autocorrelations and partial autocorrelations of the squared residuals up to any specified number of lags and computes the Ljung-Box Q-statistics for the corresponding lags. The correlograms of the squared residuals can be used to check autoregressive conditional heteroskedasticity (ARCH) in the residuals; see also ARCH LM Test below. If there is no ARCH in the residuals, the autocorrelations and partial autocorrelations should be zero at all lags and the Q-statistics should not be significant; see Series Views, Correlogram for a discussion of the correlograms and Q-statistics. This view is available for equations estimated by least squares, two-stage least squares, and nonlinear least squares estimation. In calculating the probability for Q-statistics, the degrees of freedom are adjusted for the inclusion of ARMA terms. To display the correlograms and Q-statistics of the squared residuals, push View/Residual Tests/Correlogram Squared Residuals on the equation toolbar. In the Lag Specification dialog box that opens, specify the number of lags over which to compute the correlograms.

Histogram and Normality Test This view displays a histogram and descriptive statistics of the residuals, including the Jarque-Bera statistic for testing normality. If the residuals are normally distributed, the histogram should be bell-shaped and the Jarque-Bera statistic should not be significant; see Series Views, Jarque-Bera test for further details. This view is available for residuals from least squares, two-stage least squares, nonlinear least squares, and binary, ordered, censored, and count models.

To display the histogram and Jarque-Bera statistic, select View/Residual Tests/Histogram-Normality. The Jarque-Bera statistic has a distribution with two degrees of freedom under the null hypothesis of normally distributed errors.

Serial Correlation LM Test This test is an alternative to the Q-statistics for testing serial correlation. The test belongs to the class of asymptotic (large sample) tests known as Lagrange multiplier (LM) tests. Unlike the Durbin-Watson statistic for AR(1) errors, the LM test may be used to test for higher order ARMA errors, and is applicable whether or not there are lagged dependent variables. Therefore, we recommend its use whenever you are concerned with the possibility that your errors exhibit autocorrelation. The null hypothesis of the LM test is that there is no serial correlation up to lag order p, where p is a pre-specified integer. The local alternative is ARMA(r,q) errors, where the number of lag terms p = max{r,q}. Note that the alternative includes both AR(p) and MA(p) error processes, and that the test may have power against a variety of autocorrelation structures. See Godfrey (1988) for a discussion. The test statistic is computed by an auxiliary regression as follows: suppose you have estimated the regression

where e are the residuals. The test statistic for lag order p is based on the regression . This is a regression of the residuals on the original regressors X and lagged residuals up to order p. EViews reports two test statistics from this test regression. The F-statistic is an omitted variable test for the joint significance of all lagged residuals. Because the omitted variables are residuals and not inde...