Econ 427 lecture 18 slides

1. Econ 427 lecture 18 slides Multivariate Modeling (cntd)

2. Multivariate Forecasting • Last time, we began discussing models where the evolution of one variable is related to developments in others • This is suggested by much of our theory, and exploiting this can improve forecasts-multivariate regression modeling.

3. Bivariate Regression model • where x helps determine (cause) y.

4. Distributed lag models • We saw that the most obvious way to capture cross-variable dynamics is to use a sequence of lags of the other variable • The deltas are the “lag weights” and their pattern is called the lag distribution

5. Estimation problems • Own-variable dynamics will usually also be important. We looked at different ways to handle that • lag dependent variables. • include ARMA disturbance

6. Transfer function models • Or both: the transfer function model is the most general multivariate model and includes both types of influences

7. Single or system equation models? • In some cases it may make sense to assume that right-hand side variables can be treated as exogenous for the purposes of modeling/forecasting a left-hand side variable • E.g. an individual firms revenues may depend on GDP, but not visa versa. • Formally what is needed for estimation (forecasting) is that the RHS var is weakly (strongly) exogenous with respect to the parameters we are trying to estimate.

8. Exogeneity concepts • A variable is weakly exogenous for parameters of interest if the marginal process for the variable contains no information useful for estimating those parameters • The marginal process is very roughly speaking the distribution of the variable itself without regard to particular values of variables it may be correlated with. • This means we can estimate the parameters without having to worry about the random process behind the weakly exogenous right-hand-side variable. • A variable is strongly exogenous if it is weakly exogenous AND it is not affected by lagged values of the endogenous variable. • In this case, we can forecast without having to worry about how our left-hand-side variable might affect future values of our right-hand-side variable.

9. Single or system equation models? • Often, though, we want to allow for influences running potentially in both (all) directions • System modeling approaches

10. Vector Autoregressions (VAR)

11. A VAR(1) model • A VAR(1) for a system of N=2 variables runs 2 equations where in each case 1 lags of the own and other variables are included. • where

12. A VAR(1) model • So innovations can be correlated across regressions. • If exactly the same vars are on RHS (as in this case) then OLS on individual equations can be used; otherwise must use SUR. • SIC and AIC for the complete system can be constructed.

13. A VAR(p) model • A VAR(p) for a system of N variables runs N equations where in each case p lags of the own and other variables are included. • Ex: VAR(2) for a bivariate (N=2) system: • (Here, I am just using superscripts to keep track of the lags, not to indicate powers, so there are 8 distinct parameters in this model—maybe not great notation!)

14. Analyzing dependence in a VAR system • (Granger) Causality tests. (Book calls it predictive causality) For a two variable system, where we have reason to think any influence works thru in three periods or less, we would run: • (This notation is probably easier to make sense of. I am using alphas to represent coefs on “own” lags and betas to represent coefs on lags of the other variable.)

15. Granger Causality • To test whether y2 “Granger causes” y1, we test whether • Similar for the other direction. • Is this the same meaning as our casual use of the word “causation”? • Obviously not. Here it means “are lags of y2 useful in predicting y1?”

16. Impulse-response functions • We use impulse-response functions to see how a shock to the variables affect each other. • We want to know how an innovation in one of the variables will affect itself over time and the other variable(s). • Next time…