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Chapter 10: Multicollinearity Iris Wang iris.wang@kau.se

Econometric problems

Multicollinearity What does it mean? A high degree of correlation amongst the explanatory variables What are its consequences? It may be difficult to separate out the effects of the individual regressors. Standard errors may be overestimated and t values depressed. Note: a symptom may be high R 2 but low t values How can you detect the problem? Examine the correlation matrix of regressors also carry out auxiliary regressions amongst the regressors. Look at the Variance inflating factor (VIF) NOTE: be careful not to apply t tests mechanically without checking for multicollinearity multicollinearity is a data problem, not a misspecification problem

Variance inflating inflating factor (VIF) Multicollinearity inflates the variance of an estimator VIF 2 J = 1/(1 R J2 ) where R J2 measures the R 2 from a regression of X j on the other X variable/s ibl/ serious multicollinearity problem if VIF J >5

Econometric problems

Heteroskedasticity What does it mean? The variance of the error term is not constant t What are its consequences? The least squares results are no longer efficient i and t tests and F tests results may be misleading How can you detect dt tthe problem? Plot the residuals against each of the regressors or use one of the more formal tests How can we remedy theproblem? Respecify the model look for other missing variables; perhaps take logs or choose some other appropriate functional form; or make sure relevant variables are expressed per capita

The Homoskedastic Case

The Heteroskedastic Case

The consequences of heteroskedasticity OLS estimators are still unbiased (unless there are also omitted variables) ibl However OLS estimators are no longer efficient or minimum variance The formulae used to estimate the coefficient standard errors are no longer correct so the t-tests will be misleading confidence intervals based on these standard errors will be wrong

Detecting heteroskedasticity Visual inspection of scatter diagram or the residuals Goldfeld Quandt test suitable for a simple form of heteroskedasticity

Goldfeld Quandt test (JASA, 1965) P. 382, Suppose it looks as if σ ui = σ u X i i.e. the error variance is proportional to the square of one of the X s Rank the data according to the variable and conduct an F test using RSS 2 /RSS 1 where these RSS are based on regressions using the first and last [n c]/2 observations [c is a central section of data usually about 25% of n] Reject H 0 of homoskedasticity if F cal > F tables

Remedies Respecification of the model Include relevant omitted variable(s) Express model in log-linear form or some other appropriate functional form Express variables in per capita form Where respecification won t solve the problem use robust Heteroskedastic Consistent Standard Errors (due to Hal White, Econometrica 1980)

Basic Econometrics, Spring 2012 Chapter 11: Heteroskedasticity Iris Wang iris.wang@kau.se 1

Chapter 11: Heteroskedasticity Definition: Heteroskedasticity occurs when the constant variance assumption, i.e. Var(u i X i )= σ 2, fails. This happens when variance of the error term (u i ) changes across different values of X i. Example: Savings i =α 0 +α 1 income+u i Heteroskedasticity is present if the variance of unobserved factors affecting savings (u i ) increases with income - Higher variance of u i for higher income 2

Chapter 11: Ht Heteroskedasticity kd tiit Outline 1. Consequences of Heteroskedasticity 2. Testing for Heteroskedasticity 3

1. Consequences of Heteroskedasticity OLS is unbiased and consistent under the following 4 assumptions: Linear in parameters Random sampling No perfect collinearity Zero conditional mean (E(u X)=0) Homoskedasticity assumption (MLR.4) stating constant error variance (Var(u X)= σ 2 ) plays no role in showing that OLS is unbiased & consistent Heteroskedasticity doesn t cause bias or inconsistency in OLS estimators 4

1. Consequences of Heteroskedasticity cntd However, estimators of variances, Var(β j) are biased without homoskedasticity OLS standard errors are biased Standard confidence interval, t, and F statistics which are based on standard errors are no longer valid. t & F statistics no longer have t & F distribution resp. And this is not resolved in large samples OLS is no longer BLUE and asymptotically yefficient It is possible to find estimates that are more efficient than OLS (e.g. GLS, Generalized Least Squares) Solutions involve using: i. Generalized least squares (GLS) ii. Weighted least squares (WLS) is a special case of GLS, p.373 5

Weighted Least Squares (WLS) Aim: to specify the form of heteroskedasticity detected and use weighted least squares estimator. If we have correctly specified the form of the variance, then WLS is more efficient than OLS If we used wrong form of variance, WLS will be biased but tit is generally consistent as long as E(u X)=0. But, efficiency of WLS is not guaranteed when using wrong form of variance. We use this to transform the original regression equation with homoskedastic error term i.e. the bias will improve with large N 6

2. Testing for Heteroskedasticity Why test for heteroskedasticity? First, unless there is evidence of heteroskedasticity, many prefer to use the usual t under OLS This is because the usual t statistics have exact t distribution under the assumptions of homoskedasticity & normally distributed errors. Second, if heteroskedasticity is present, it is possible to obtain better estimator than OLS when the form of heteroskedasticity is known. In the regression model: Y= β 0 +β 1 x 1 + +β k x k +u We assume that E(u x 1, x k )=0 OLS is unbiased and consistent. In order to test for violation of the homoskedasticity assumption, we want to test the null hypothesis: Ho: Var(u x 1,, x k )=σ 2 7

2. Testing for Heteroskedasticity cntd To test the null hypothesis above, we test whether expected value of u 2 is related to one or more of the explanatory variables. If we reject Ho, then heteroskedasticity is a problem & needs to be solved. Two types heteroskedasticity tests: A. Goldfeld Quandt Test for heteroskedasticity, p.382 B. White s General Heteroskedasticity kd ii Test, p.386 Once we reject Ho of homoskedasticity, we should treat the heteroskedasticity problem 8

B. White heteroskedasticity test homoskedasticity assumption, Var(u X)=σ 2, can be replaced with weaker assumption that u 2 is uncorrelated with: All the independent variables (x j ) Their squared terms (x 2 j) and Their cross products (x j x h for all h j) Under this weaker assumption, OLS standard errors and test statistics are asymptotically valid Whiteheteroskedasticity heteroskedasticity test is motivatedby thisassumption assumption. For e.g. for k=3, û 2= δ 0 + δ 1 x 1 + δ 2 x 2 + δ 3 x 3 +δ 4 x 2 1 + δ 5 x 2 2 + δ 6 x 2 3+ δ 7 x 1 x 2 + δ 8 x 1 x 3 + δ 9 x 2 x 3 +v White test is F statistics for testing all δ j, except δ 0,arezero. Limitation: it consumes degrees of freedom (for k=3, we needed 9 variables) 9

Basic Econometrics Autocorrelation Iris Wang iris.wang@kau.se

Econometric problems

Topics to be covered Overview of autocorrelation First order autocorrelation and the Durbin Watson test Higher order autocorrelation and the Breusch Godfrey test Dealing with autocorrelation Examples and practical illustrations

Autocorrelated series and autocorrelated Autocorrelated series and autocorrelated disturbances

Overview of autocorrelation What is meant by autocorrelation? The error terms are not independent from observation to observation u t depends ds on one eor more epast values uesof u What are its consequences? The least squares estimators are no longer efficient (i.e. they don t have the lowest variance). More seriously autocorrelation may be a symptom of model misspecification ifi How can you detect the problem? Plot the residuals against time or their own lagged values, calculate the Durbin Watson statistic or use some other tests of autocorrelation such as the Breusch Godfrey (BG) test How can you remedy the problem? Consider possible model re specification of the model: a different functional form, missing variables, lags etc. If all elsefails you couldcorrectcorrect for autocorrelation by using the Cochrane Orcutt procedure or Autoregressive Least Squares

First order autocorrelation

The sources of autocorrelation

The consequences of autocorrelation

Detecting autocorrelation

The Durbin Watson test