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Regression Analysis. Introduction. Derive the α and β Assess the use of the T-statistic Discuss the importance of the Gauss-Markov assumptions Describe the problems associated with autocorrelation, how to measure it and possible remedies Introduce the problem of heteroskedasicity.
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Introduction • Derive the α and β • Assess the use of the T-statistic • Discuss the importance of the Gauss-Markov assumptions • Describe the problems associated with autocorrelation, how to measure it and possible remedies • Introduce the problem of heteroskedasicity
Deriving the α and β • The aim of a least squares regression is to minimize the distance between the regression line and error terms (e).
T-test • When conducting a t-test, we can use either a 1 or 2 tailed test, depending on the hypothesis • We usually use a 2 tailed test, in this case our alternative hypothesis is that our variable does not equal 0. In a one tailed test we would stipulate whether it was greater than or less than 0. • Thus the critical value for a 2 tailed test at the 5% level of significance is the same as the critical value for a 1 tailed test at the 2.5% level of significance.
T-test • We can also test whether our coefficient equals 1.
Gauss-Markov Assumptions • There are 4 assumptions relating to the error term. • The first is that the expected value of the error term is zero • The second is that the error terms are not correlated • The third is that the error term has a constant variance • The fourth is that the error term and explanatory variable are not correlated.
Gauss-Markov assumptions • More formally we can write them as:
Additional Assumptions • There are a number of additional assumptions such as normality of the error term and n (number of observations) exceeding k (the number of parameters). • If these assumptions hold, we say the estimator is BLUE
BLUE • Best or minimum variance • Linear or straight line • Unbiased or the estimator is accurate on average over a large number of samples. • Estimator
Consequences of BLUE • If the estimator is not BLUE, there are serious implications for the regression, in particular we can not rely on the t-tests. • In this case we need to find a remedy for the problem.
Autocorrelation • Autocorrelation occurs when the second Gauss-Markov assumption fails. • It is often caused by an omitted variable • In the presence of autocorrelation the estimator is not longer Best, although it is still unbiased. Therefore the estimator is not BLUE.
Durbin-Watson Test • This tests for 1st order autocorrelation only • In this case the autocorrelation follows the first-order autoregressive process
Zone of indecision Zone of indecision 0 dl du 2 4-du b-dl 4 Durbin-Watson Test- decision framework
DW Statistic • The DW test statistic lies between 0 and 4, if it lies below the dl point, we have positive autocorrelation. If it lies between du and 4-du, we have no autocorrelation and if above 4-dl we have negative autocorrelation. • The dl and du value can be found in the DW d-statistic tables (at the back of most text books)
Lagrange Multiplier (LM) Statistic • Tests for higher order autocorrelation • The test involves estimating the model and obtaining the error term . • Then run a second regression of the error term on lags of itself and the explanatory variable: (the number of lags depends on the order of the autocorrelation, i.e. second order)
LM Test • The test statistic is the number of observations multiplied by the R-squared statistic. • It follows a chi-squared distribution, the degrees of freedom are equal to the order of autocorrelation tested for (2 in this case) • The null hypothesis is no autocorrelation, if the test statistic exceeds the critical value, reject the null and therefore we have autocorrelation.
Remedies for Autocorrelation • There are 2 main remedies: • The Cochrane-Orcutt iterative process • An unrestricted version of the above process
Heteroskedasticity • This occurs when the variance of the error term is not constant • Again the estimator is not BLUE, although it is still unbisased it is no longer Best • It often occurs when the values of the variables vary substantially in different observations, i.e. GDP in Cuba and the USA.
Conclusion • The residual or error term is the difference between the fitted value and actual value of the dependent variable. • There are 4 Gauss-Markov assumptions, which must be satisfied if the estimator is to be BLUE • Autocorrelation is a serious problem and needs to be remedied • The DW statistic can be used to test for the presence of 1st order autocorrelation, the LM statistic for higher order autocorrelation.