Introduction to Econometrics

1. Introduction to Econometrics

3. Yt = a + b Xt-1 + ut Changes in X affect Y but with a known lag (in this case one period). Provided the length of the lag is known, or is easily established, this raises no new problems. Indeed it can be helpful from a forecasting point of view because the value of the independent variable will be known with certainty at the time when the next forecast of Y is to be made. EXAMPLE: Forecasting employment in Orange County (California) EMPt = a + b RGNPt + ut where EMPt denotes total employment in the county in quarter t, RGNPt-1denotes real GNP for the whole of the US in the previous quarter. Source: Doti and Adibi (1998) Here we can actually exploit the lags in the relationship for forecasting purposes

4. Yt = a + b0Xt + b1 Xt-1 +�.+ bsXt-s+ ut where s is the maximum lag allowed for. Rather than assume that the whole of the affect is delayed, this model has the effect distributed over a number of periods. Problems: establishing the maximum lag s loss of degrees of freedom possible multicollinearity Example. Accidents and safety training; Koop (2000) Yt = a + b0Xt + b1 Xt-1 +..+ b4Xt-4+ ut where Yt = losses due to accidents for a company (�/month) Xt = hours of safety training provided to each worker in month t A simple regression of Y on X appeared to show no relationship between these variables - although the DW stat suggested misspecification.

5. Suppose that we anticipate a gradual decline in the affect of X on Y as the number of periods increase. For example Y might be sales and X advertising expenditure. If we can assume a geometric rate of decline and an infinite lag structure we can use the Koyck transformation to produce a simple model with just Xt and Yt-1 as regressors Writing Yt = a + b0Xt + b1 Xt-1 +�.+ bsXt-s +�..+ ut [1] If bj+1/bj = ? for all j (with b0 just = b) [1] becomes Yt = a + bXt + ?bXt-1 +�.+ ?sbXt-s + ...�.+ ut [2] Lag [2] by one period and multiply by ? ? Yt-1 = ? a + ? bXt-1+ ? 2bXt-2 +�.+ ? sbXt-s + �.+ ? ut [3] Subtract [3] from [2] and rearrange Yt = a(1- ?) + bXt + ? Yt-1 + ut - ? ut-1 [4]

13. Robert E Hall (JPE 1978) suggested that consumption would follow a simple first-order autoregressive process if (1) consumption depends only upon permanent income (YP) (2) agents� expectations are formed rationally The second assumption means that YPt = YPt-1 + ?t where E(?t) = 0 ?t represents the revision made to agents� perceived permanent income in period t. Individuals out not to expect their permanent income to change � if they did this knowledge should already have been used to reassess permanent income � so Hall�s consumption function is sometimes known as the �surprise� consumption function � ?t is the surprise. (1) requires Ct = K YPt Substituting we find that Ct = Ct-1 + K ?t or Ct = Ct-1 + et Consumption should follow a random walk. The model is easily testable against less restricted models and is typically rejected.

15. Begin with a general model which nests the restricted model and so allows any restrictions to be tested These restrictions may be suggested either by theory � or by empirical results

16. TEST 1 First ensure that the general model does not suffer from any diagnostic problems. Examine the residuals in the general model to ensure that they possess acceptable properties. (Test for problems of autocorrelation, heteroskedasticity, non-normality, incorrect functional form etc.)

17. General to specific modelling TEST 2 Now test the restrictions implied by the specific model against the general model � either by exclusion tests or other tests of linear restrictions.

18. General to specific modelling TEST 3 If the restricted model is accepted, test its residuals to ensure that this more specific model is still acceptable on diagnostic grounds

20. �Should I include all the variables in the database in my model?� �How many explanatory variables do I need in my model?� �How many models do I need to estimate?� �What functional form should I be using?� �Do I need to include lagged variables?� �What are interactive dummies � do I need them?� �Which regression model will work best and how do I arrive at it?�

21. Maybe several hundred observations Maybe 10-12 potential explanatory variables, some of which will be dummy variables. So plenty of degrees of freedom but still lots of potential models to try, especially if you consider alternative functional forms, interactive dummies Maybe problems of multicollinearity, heteroskedasticity and non-normality Model selection is not just a matter of maximizing Rbar-squared over all possible models (or some other criterion) Use economic theory and past studies to identify �core� variables Test exclusion restrictions from a general model but balanced against misspecification tests. �Informed� searches.

22. Maybe only around a hundred observations Maybe four or five potential explanatory variables, some of which may be dummy variables. Relatively few degrees of freedom but still lots of potential models to try, especially if you consider alternative functional forms, lagged variables and interactive dummies As well as problems of multicollinearity, heteroskedasticity and non-normality there may be issues of autocorrelation and non-stationarity Model selection is not just a matter of maximizing Rbar-squared over all possible models Use economic theory and past studies to identify �core� variables and if possible functional form Test exclusion and other restrictions from a general model but balanced against misspecification tests. �Informed� searches.