2Introduction Derive the α and β Assess the use of the T-statistic Discuss the importance of the Gauss-Markov assumptionsDescribe the problems associated with autocorrelation, how to measure it and possible remediesIntroduce the problem of heteroskedasicity
7T-testWhen conducting a t-test, we can use either a 1 or 2 tailed test, depending on the hypothesisWe 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.
8T-testWe can also test whether our coefficient equals 1.
9Gauss-Markov Assumptions There are 4 assumptions relating to the error term.The first is that the expected value of the error term is zeroThe second is that the error terms are not correlatedThe third is that the error term has a constant varianceThe fourth is that the error term and explanatory variable are not correlated.
10Gauss-Markov assumptions More formally we can write them as:
11Additional 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
12BLUE Best or minimum variance Linear or straight line Unbiased or the estimator is accurate on average over a large number of samples.Estimator
13Consequences of BLUEIf 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.
14AutocorrelationAutocorrelation occurs when the second Gauss-Markov assumption fails.It is often caused by an omitted variableIn the presence of autocorrelation the estimator is not longer Best, although it is still unbiased. Therefore the estimator is not BLUE.
15Durbin-Watson Test This tests for 1st order autocorrelation only In this case the autocorrelation follows the first-order autoregressive process
16Durbin-Watson Test- decision framework dldu244-dub-dlZone of indecision
17DW StatisticThe 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)
18Lagrange Multiplier (LM) Statistic Tests for higher order autocorrelationThe 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)
19LM TestThe 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.
20Remedies for Autocorrelation There are 2 main remedies:The Cochrane-Orcutt iterative processAn unrestricted version of the above process
21HeteroskedasticityThis occurs when the variance of the error term is not constantAgain the estimator is not BLUE, although it is still unbisased it is no longer BestIt often occurs when the values of the variables vary substantially in different observations, i.e. GDP in Cuba and the USA.
22ConclusionThe 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 BLUEAutocorrelation is a serious problem and needs to be remediedThe DW statistic can be used to test for the presence of 1st order autocorrelation, the LM statistic for higher order autocorrelation.