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1、Financial EconometricsTutorial 2 SolutionDr Imran ShahUniversity of Bath12TerminologyRegression ModelYi = a + bxi + uiy is thexs are the predictandpredictorsregressandregressorsexplained variableexplanatory variablesdependent variableindependent variablesendogenous variableexogenous variablesleft ha

2、nd side variableright hand side variablesEconomics 20 - Prof. Anderson3Ordinary Least Squares Basic idea of regression is to estimate the population parameters from a sample. Let (xi,yi): i=1, ,n denote a random sample of size n from the population. For each observation in this sample, it will be th

3、e case that yi = b0 + b1xi + uiOrdinary Least Squares a regression estimation technique that calculates the Beta-hats - estimated parameters or coefficients of the model so as to minimize the sum of the squared residuals.Classical Assumption 1The error term has a zero population mean.We impose this

4、assumption via the constant term.The constant term equals the fixed portion of the Y that cannot be explained by the independent variables.The error term equals the stochastic portion of the unexplained value of Y.Classical Assumption 2The error term has a constant varianceHeteroskedasticityWhere do

5、es this most often occur? Cross-sectional dataWhy does this occur in cross-sectional data?Classical Assumption 3Observations of the error term are uncorrelated with each other.Serial Correlation or AutocorrelationWhere does this most often occur? Time-series dataWhy does this occur in time-series da

6、ta?Classical Assumption 4-5The data for the dependent variable and independent variable(s) do not have significant measurement errors.The regression model is linear in the coefficients, is correctly specified, and has an additive error term.Classical Assumption 6The error term is normally distribute

7、dThis is an optional assumption, but a good idea. Why?One cannot use the t-statistic or F-statistic unless this holds (will explain these later).Five More AssumptionsAll explanatory variables are uncorrelated with the error term.When would this not be the case? Then a system of equations is needed (

8、i.e. supply and demand).What are the consequences? Estimation of slope coefficient for correlated X terms is biased.No explanatory variable is a perfect linear function of any other explanatory variable.Perfect collinearity or multicollinearityConsequence: OLS cannot distinguish the impact of each X

9、 on Y.X values are fixed in repeated sampling.The number of observations n must be greater than the number of parameters to be estimated. There must be variability in X and Y values. Topic Nine Serial Correlation 1.Introduction Autocorrelation Autocorrelation occurs in time-series studies when the e

10、rrors associated with a given time period carry over into future time periods. For example, if we are predicting the growth of stock dividends, an overestimate in one year is likely to lead to overestimates in succeeding years.Topic Nine Serial Correlation 1.Introduction Times series data follow a n

11、atural ordering over time. It is likely that such data exhibit intercorrelation, especially if the time interval between successive observations is short, such as weeks or days.Topic Nine Serial Correlation 1.Introduction We expect stock market prices to move or move down for several days in success

12、ion.In situation like this, the assumption of no auto or serial correlation in the error term that underlies the CLRM will be violated. We experience autocorrelation when Consequences of Using OLS OLS Estimation Allowing for Autocorrelation As noted, the estimator is no more not BLUE, and even if we

13、 use the variance, the confidence intervals derived from there are likely to be wider than those based on the GLS procedure. Hypothesis testing: we are likely to declare a coefficient statistically insignificant even though in fact it may be. One should use GLS and not OLS. 13 Consequences of Using

14、OLS OLS Estimation Disregarding Autocorrelation The estimated variance of the error is likely to overestimate the true variance Over estimate R-squareTherefore, the usual t and F tests of significance are no longer valid, and if applied, are likely to give seriously misleading conclusions about the

15、statistical significance of the estimated regression coefficients. 14 Causes of Autocorrelation Inertia - Macroeconomics data experience cycles/business cycles. Specification Bias- Excluded variableAppropriate equation: Estimated equation Estimating the second equation implies 15 Causes of Autocorre

16、lation Specification Bias- Incorrect Functional Form 161718Durbin-Watson testWe can also do a formal hypothesis test, (the Durbin-Watson test) for independenceThe test assumes the errors follow a model of the form19Where the uis are independent, normal and have constant variance. r is the lag 1 corr

17、elation: this is the autoregressive model of order 1NBDurbin-Watson test (2)When r = 0, the errors are independentThe DW test tests independence by testing r = 0r is estimated by 20Durbin-Watson test (3)DW test statistic is 21Value of DW is between 0 and 4Values of DW around 2 are consistent with in

18、dependenceValues close to 4 indicate negative serial correlationValues close to 0 indicate positive serial correlationDurbin-Watson test (4)8/13/2022330 lecture 1222There exist values dL, dU depending on the number of variables k in the regression and the sample size n see table on next slideUse the

19、 value of DW to decide on independence as follows:044-dU4-dLdLdUPositive autocorrelationNegative autocorrelationIndependenceInconclusive8/13/2022330 lecture 122324(ii): Number of observations: n=45.Explanatory variable =1.DL= 1.48 and DU= 1.57.Our Durbin Watson statistic (DW=1.52) lies between these

20、 2 values and so is in the zone of indecision, we dont know if we have autocorrelation or not.The Breusch-Godfrey TestThe Breusch-Godfrey (LM) TestThe Breusch-Godfrey Test28Cochrane-Orcutt methodologyQuestion 3:This approach to remedying autocorrelation relies on the generalised difference equation

21、approachIt is an iterative procedure, the estimates of the parameters converge towards their actual value.It assumes the autocorrelation follows a first order autoregressive process (simplest form).29Cochrane-Orcutt methodologyProcedure: Run regression and collect the error term (u).Run a second reg

22、ression of u against ut-1 to obtain an estimate of .Form the generalised difference equation, to obtain new estimates of the parametersRe-run this equation, obtain u and repeat process until there is no significant difference from one iteration to the next.30Cochrane-Orcutt methodologyRestricted Ver

23、sion of the GDEThe generalised difference equation can be written in the following form, which includes a restriction, the product of the coefficients on y(-1) and x is equal to the negative of the coefficient on x(-1) :The Common Factor Test To determine whether the restricted or unrestricted versi

24、ons are best, we can use the common factor test, under the null hypothesis that the restriction holds, the test statistic is:31Question 3:RSSR=0.79 and RSSUR=0.56T=Observations =32Cochrane-Orcutt methodologyCommon Factor =We lose one observation when making the logs.Critical Value is chi-squared (1)

25、 distribution=3.84As calculate value is greater than table value, we reject the hypothesis that the restricted version (Cochrane-Orcutt) is the best method to e the autocorrelation and therefore the unrestricted version is best.32HeteroskedasticityQuestion 4:For example, consider a regression of hou

26、sing expenditures on e.Consumers with low values of e have little scope for varying their rent expenditures. Var(ei ) is low.Wealthy consumers can choose to spend a lot of money on rent, or to spend less, depending on tastes. Var(ei ) is high.33HeteroskedasticityWhat are the implications of heterosk

27、edasticity for OLS?Under the GaussMarkov assumptions (including homoskedasticity), OLS was the Best Linear Unbiased Estimator.Under heteroskedasticity, is OLS still Unbiased?Is OLS still Best?Implications of Heteroskedasticity:OLS is still unbiased.OLS is no longer efficient; some other linear estim

28、ator will have a lower variance.Estimated Standard Errors will be incorrect; C.I.s and hypothesis tests (both t- and F- tests) will be incorrect.34HeteroskedasticityQuestion 5: White TestFive Steps of the White Test:Regress Y against your various explanators using OLSCompute the OLS residuals, e1.en

29、Regress ei2 against a constant, all of the explanators, the squares of the explanators, and all possible interactions between the explanators (p slopes total)Five Steps of the White Test (cont.)Compute R2 from the “auxilliary equation” in step 3Compare nR2 to the critical value from the Chi-squared

30、distribution with p degrees of freedom.35HeteroskedasticityQuestion 5 (i): White TestConclusion: As 8.75.99 at 5%. We reject the null hypothesis of no heteroskedasticity. (ii). Consider the following model:Yt = 1+ Xt+ ut36HeteroskedasticityQuestion 5 (i): White TestRunning the above regression remov

31、es the heteroskedasticity as the error term now has a constant variance.37ParsimoniousExercise 4: Question 1In general it is better to have a parsimonious model, the model with only the most relevant variables), rather than a large model. This is because a small model with just a few relevant is eas

32、ier to interpret and usually give better forecasts. Also it reduces the chances of multicollinearnity (two explanatory variables being correlated). However the disadvantage of a small model is that you may have left out an important explanatory variable. This can be serious and produces omitted vari

33、able bias. So in general we try to include all significant variables (5% level of significance) but leave out all others. 38R-SquaredExercise 4: Question 2The R2 statistic, which measure the explanatory power of a regression, is not always appropriate in a multiple regression because it will always

34、rise in value when an additional variable is added to the regression, regardless of whether the variable is significant. So if we sought to maximise the R2 statistic as a means of selecting the best model, we would always chose the one with the largest number of explanatory variables, even though mo

35、st are insignificant. 39While R2 rises with the number of explanatory variablesDefine another goodness-of-fit measure: called Adjusted R2Adjusted R-squaredThe adjusted R2 statistic takes into account the extra variables when deciding on the explanatory power of the regression. So it can fall when ex

36、tra variables are added. In general if the t-statistic exceeds 1, adding an extra variable increases the adjusted R2 statistic, so it may rise even though the extra variable is not significant (i.e. above 2 at the 5% level of significant).decreases with the number of explanatory variables (k)imposes

37、 a penalty for adding additional explanatory variablesR-SquaredExercise 4: Question 2Lecture 1440F-stat in terms of R2The F-test for the joint hypothesis: H0: b1 = b2 = 0 (All the explanatory variables are jointly equal to zero)H0: b1 b2 (All the explanatory variables are jointly not equal to zero)F-stat can be written in terms of R2:Recalling our question output, we can substitute R2 = 0.36, n=50 and k=3.Critical value F(2,47)= 3.23. As 13.24 3.23 We would reject the null at