Difference Between The Regression Coefficients - Essay Example

Chow test Chow test is used to find the significant difference between the regression coefficients of two sub samples. “The Chow test is used to examine the effect of structural change due to price or policy alteration on a regression model involving time series data” (Gujarati, 2003). If there are two sub-samples (say, based on two time periods) we find the Residual Sum of Squares for each sample. Let the Residual Sum of Squares of the first sample be RSS1, Residual Sum of Squares of the second sample be RSS2. Now let the Residual Sum of Squares be RSS. Then the F ratio is defined as ((RSS – RSS1 – RSS2 )/k) / ((RSS1+RSS2+…+RSSk)/(n1+n2-2k)) which follows F distribution with k, n1+n2 - 2k df. The homoscedasticity of errors in Chow test is assumed to be unique which means that the disturbance terms in both the sub samples are the same. The structural changes may be caused by differences in the intercept or slope or both. To find this Chow test is more useful. Here the assumption is, in both the samples error terms follow Normal distribution with mean 0 and common (homoscedastic) variance 2. The two error terms are independently distributed. The Chow test supports savings-income relationship that has undergone structural change over a long period. This test will be useful only if the two regressions are different without paying attention to the intercepts. Under the Null hypothesis of equality of variances in the two sub-populations, first of all we must ensure that the two variances do not show any significant difference and then only go for Chow test. If the variances do not show significant differences, the Chow test should not be used in the case. However if the error variances in the two sub-populations are heteroscedastics, the Chow test can be modified. Also the Chow test seems to be more sensitive to the choice of time at which the regression parameters have changed. Suppose we want to test the auto regression of the dependent variable expenditure on the independent variable income we have to run the procedure REGRESS in STATA. Let us proceed as follows with two time periods with dependent variable expenditure (exp) and independent variable income (income): . bysort timeper: regress exp income . regress exp income if timeper (time period 0) for eg.1987 to 1996 . regress exp income if !timeper (time period 1) for eg.1997 to 2006 To combine both in the constrained model, Expenditure is regressed on income for both time periods together as follow: . reg exp income Null hypothesis: The regression coefficients do not differ significantly for the two time periods time period 0 and time period 1. Alternative hypothesis: The regression coefficients differ significantly for the two time periods time period 0 and time period 1. Chow test is being considered as an important tool in predicting the differences between the two regression coefficients when the groups are heterogeneously different. Even with the impact of dummy variables Chow test seems to be more robust in predicting the significant difference between two regressions under homoscedastic situations. However with some slight modifications Chow test is applicable in heteroscedastic situations. Breusch – Pagan test Breusch – Pagan test is used in testing whether heteroscedasticity is present in a linear regression model. It is also used to test whether the estimated variance of the residuals from regression equation are dependent on the values of independent variable values. “Breusch-Pagan test will be performed in order to identify any heteroskedasticity that is present in the models. The Breusch-Pagan test specifically tests for heteroskedasticity related to the independent variables and it must be assumed that the error term is normally distributed” (Gujarati, 2003). Consider the following regression model with k variables X1, X2, X3, …. Xk Let the regression model with the k variables be Yi = 1+ 2 X2i +…….+ k Xki + ui Let the error variance be i2 which is a function of Z variables i2 = f(1 + 2 Z2i + ….+ m Zmi) If Z variables are held constant over time, then i2 becomes i2 = 1 + 2 Z2i + ….+ m Zmi ie. i2 is a function of Z’s. If all ’s are equal ie. 2 = 3 =….= m = 0 then i2 = 1. Hence to test whether i2 is homosdcedastic, we can formulate the hypothesis that 2 = 3 =….= m = 0 which is the basis of Breusch – Pagan test. The first step in the Breusch – Pagan test is to estimate Yi = 1+ 2 X2i +…….+ k Xki + ui using Ordinary Least Square method to obtain the residuals 1, 2,…. n and evaluate 2 = i2/n which is the Maximum Likelihood Estimator of 2. Then we evaluate pi’s which are given as pi = i2/2 ie. Each residual is first squared and then divided by 2. Now we write the regression equation with pi as dependent variable and Z’s as independent variable as pi = 1 + 2 Z2i + ….+ m Zmi + vi where vi is the residual term in the above regression equation, then we obtain the Explained Sum of Squares as  = ESS / 2, on the assumption that ui’s are normally distributed we can show that if homoscedasticity is present and if the sample size n can be indefinitely increased then  asymptotically follows Chi – Square distribution with m - 1 degrees of freedom,. Hence if this  is exceeding the critical value of Chi Square (with m – 1 df.) at the specified level of significance, then we can reject the hypothesis and say that homoscedasticity is not present. We also ensure that the auto correlation is not present among the independent variables. A Durbin Watson test also confirms that there may not be auto correlation among the variables. Here we assume that 1. Particularly, the independent variables are not stochastic in nature; 2. The error terms follow normal distribution and 3. The regression equation does not involve lagged values of the regressed variables and to ensure this Durbin Watson statistical test has to be adopted. Breusch – Pagan test using the STATA package: . bpagan exp income The Null Hypothesis is H0: The variance is constant / The independent variable is homoscedastic in nature. Feasible generalized least squares Feasible Generalized Least Squares (FGLS) is a regression procedure similar to the technique of generalized least squares with only the difference is that it utilizes the estimated variance - covariance matrix because o the nature of the true matrix is not known in advance. In Feasible Generalized Least Squares the model is given by y = Xβ + uj X is the matrix of independent variables and β is a column matrix of the estimable parameters. The residual vector uj is not expected to possess the variances equal: the assumption being they have different unknown variances in spite of not being correlated among themselves. The assumption is residual vector has a diagonal covariance matrix Ω. The method of Ordinary Least Squares estimation is applicable to a linear system having heteroscedastic errors, but any how Ordinary Least Squares do not possess Best Linear Unbiased Estimators, Hence we have to find a procedure to estimate the error variance-covariance Ω with the following iterations: OLS = (X’X)-1 X’ y and estimates of the residuals j are constructed. The need to go for Feasible Generalized Least Squares is to estimate the parameters very near to BLUE for small samples without affecting the homoscedasticity. The Ordinary Least Square coefficients have the usual properties for large samples, but the Feasible Generalized Least Squares even in small samples one can maintain the Best Linear Unbiased Estimator properties as well. Here we cut off the first value in the sample and proceed. Since OLS estimators are consistent despite autocorrelation in large samples, it makes little difference whether we estimate correlation coefficient from the Durbin – Watson d statistics or from the regression of the residuals in the current period on the residuals of the previous period, or from the Cochrane – Orcutt iterative procedure because they all provide consistent estimates of the true correlation coefficient. It is assumed to be a general principle that whenever we use an estimator in place of its true value, the estimated Ordinary Least Square coefficients may have the usual optimum properties asymptotically in large samples. But in small samples, we have to be taking more care in interpreting the results estimated. Therefore even though the correlation coefficient measured from small samples the result may not be applicable to large samples and the correlation coefficient measured from small samples is too low with which we should not generalize it for large samples. Using STATA software the FGLS is run as follows: . regress exp income, noconstant . constraint 1 t1 + t2 = 0 where t1 + t2 = 0 is the constraint . cnsreg exp income t1 t2 const(1). Ramsey RESET Test Ramsey suggested a general test of specification error known as RESET (regression specification error test). In this procedure we consider the input – output model as Yi = λ1 + λ2Xi + ui -------- (1) Where Y is the dependent variable and X is the independent variable. Consider the plotting of residuals obtained from the above regression model. Here the estimated Yi ie. ‘s are plotted against the actual Yi’s. Here the Σi and uiare necessarily assumed to be zeros. Hence it is suggested that introduction of as regressor to increase R2 to make it statistically significant on the basis of variance ratio test (F test). The procedure involved in RESET procedure is as follows: 1. From the given model estimate the Yi’s ie. . 2. Now we have to introduce in some form as an additional regressor. If there is a curvilinear regression relationship between i and , then introduction of higher powers of , ie. 2 and 3 to be done as additional regressors (independent variables). Hence Yi = β1 + β2 Xi + β3 2 + β4 3 + ui -------- (2) 3. Let R2 be obtained from equation (1) be R2pre and from equation (2) be R2post and we can use the F test namely (R2post – R2pre) / Number of regressors F = ----------------------------------------------------------------------- (1 – R2post) / (n – Number of parameters in the new model) to find out the increase in R2 is significantly different. 4. If the computed F value is significant at 95% confidence level, we accept the hypothesis that the model is misspecified. Otherwise, if the F value is not significant the model not misspecified. The main advantage of RESET is that it is easily applicable because it does not require to specify what the alternative model is. But even with this information it is disadvantageous because if the model is misspecified it does not help in necessarily choosing a better alternative. Using Stata we can compare two regression runs. Specification error Simply we use either ovtest or ovtest, rhs to run the Ramsey RESET test. Here it is to be observed that both the RESET test with powers of the approved fitted values and the test with the powers of the independent variables generates a significant F test for specification error. We can improve the specification by changing specification of the model in a better manner. To begin with, we should know the impact of change of dependent variable with reference to the change in the independent variable steadily throughout the entire period. Then we find out the improvement in the R2 with pre and post regression models and find out whether R2 has improved significantly improved or not. The model has to be resolved (or at least reduced) with the specification error that the RESET test indicates is present. Ramsey RESET test has to be tested for occurrence or indication of reduction in specification error, but the specification error can not be completely avoided. But the procedure for selecting between the two versions of the test is not clearly specified. Using STATA software the Ramsey RESET test is run as follows .regress exp income It is suggested that indication of specification error is related to the functional form of variables in the model (using the powers of independent variables or the powers of fitted values of the independent variables). It is to be noted that specification error has to be considerably by the intensity of the F values, but if it is significant then we have to test with the powers of the independent variables. This clearly indicates that the functional form of the model works very fine with the inclusion of non-linear models, but still the lagged endogenous variables have to be taken care first. Reference: Gujarati, Damodar N. “Basic Econometrics”. McGraw-Hill/Irwin. Fourth edition, 2003. Read More