class: center, middle, inverse, title-slide .title[ # Econometrics - Lecture 9 ] .subtitle[ ## Additional Topics in Time Series Regression ] .author[ ### Jonas Björnerstedt ] .date[ ### 2023-11-22 ] --- ## Lecture Content - Chapter 15. Estimation of Dynamic Causal Effects - Chapter included at an overview level --- ## 15.2 Dynamic causal effects - Causal effects and time series - Two types of exogeneity, strict and sequential - Discuss strict exog (we can discuss what is in the error term) - Strict exogeneity required for unbiased estimates - AR(1) model not unbiased --- ## The distributed lag model `$$Y_t = \beta_0 + \beta_1 X_{t} + \beta_2 X_{t-1} + \beta_3 X_{t-2} + u_t$$` - Autocorrelated `\(u_t\)` - For example `$$u_t = \phi u_{t-1} + v_t$$` - Assume that `$$E\left(u_{t}\left| X_{t}, X_{t-1}, X_{t-2}\right.\right)=0$$` - Implies that `\(X\)` is not correlated with the error term - Autocorrelated `\(u_t\)` - Consistent parameter estimates - Incorrect standard errors and inference (as before) --- ## Effect of permanent change in `\(X\)` `$$Y_t = \beta_0 + \beta_1 X_{t} + \beta_2 X_{t-1} + \beta_3 X_{t-2} + u_t$$` - Assume `\(\beta_2 \neq 0\)` and `\(\beta_3 \neq 0\)` - `\(X_t\)` will have an effect on `\(Y_{t+1},Y_{t+2}\)` - Effect in `\(t+1\)` is `\(\beta_2\)` - Effect in `\(t+2\)` is `\(\beta_3\)` - What is the effect of a __permanent__ change in `\(X\)`? - Will the total effect be significant? - Confidence intervals for this effect? --- ## Dynamic multipliers - Dynamic multipliers - Effect of increase in `\(X_t\)` on `\(Y_{t+h}\)` is the `\(h\)` period _dynamic multiplier_ - Corresponds to: `\(\beta_{h+1}\)` - Cumulative dynamic multipliers - Total effect on `\(Y\)` in the next `\(h\)` periods - Sum of the effects in each period - Corresponds to: `$$\delta_h = \beta_{1} + \beta_{2} + \ldots + \beta_{h+1}$$` --- ## Cumulative dynamic multipliers `$$Y_t = \beta_0 + \beta_1 X_{t} + \beta_2 X_{t-1} + \beta_3 X_{t-2} + u_t$$` - Can estimate `\(\delta_h\)` instead of `\(\beta_h\)` directly - If cumulative multiplier is of interest - Obtain confidence intervals for cumulative effects - Express `\(X_t\)` and `\(X_{t-1}\)` as changes from `\(X_{t-2}\)` `$$X_{t-1} = (X_{t-1} - X_{t-2}) + X_{t-2} =\Delta X_{t-1} + X_{t-2}$$` `$$X_t = \Delta X_{t} +\Delta X_{t-1} + X_{t-2}$$` - We get: `$$Y_t = \beta_0 + \beta_1 (\Delta X_{t}+\Delta X_{t-1}+X_{t-2}) + \beta_2( \Delta X_{t-1}+X_{t-2}) + \beta_3 X_{t-2} + u_t$$` `$$Y_t = \beta_0 + \beta_1 \Delta X_{t}+ ( \beta_1 + \beta_2) \Delta X_{t-1} + (\beta_1 + \beta_2 +\beta_3 )X_{t-2} + u_t$$` --- ## Granger causality - `\(X_{t}\)` does not _Granger cause_ `\(Y_t\)` if - Given other regressors `\(V_{t},W_{t},\ldots\)` - `\(X_{t-1},X_{t-2},X_{t-3},\ldots\)` _all_ have zero coefficients - `\(X_{t-s}\)` do not help predict `\(Y_t\)` - Null hypothesis test: Joint F test that all coefficents are zero - Better name: _Granger predictability_ - `\(X\)` does not cause `\(Y\)` to change, but is helpful in predicting it - Observing people with umbrellas can Granger cause rainfall --- ## Forecast uncertainty Four types of error in forecasting: 1. **Specification error** - the assumptions of the model are not met 1. **Conditioning error** - the value of `\(X_{T+1}\)` on which estimate is conditioned, is uncertain 1. **Sampling error** - the estimated coefficients `\(\hat\beta_k\)` are uncertain 1. **Random error** - the realization of the error term `\(u_{T+1}\)` is unknown - Forecast intervals - Usually only include the last two types - Wider confidence intervals than 95% are common --- ## Quasi difference and GLS Rewrite `$$Y_t = \phi Y_{t-1} + (1-\phi ) \beta_0 + \beta_1 X_{t} - \phi \beta_1 X_{t-1} + v_t$$` `$$Y_t - \phi Y_{t-1} = (1-\phi ) \beta_0 + \beta_1 (X_t - \phi X_{t-1}) + v_t$$` - Define `\(\tilde{X_t} = X_t - \phi X_{t-1}\)`, `\(\tilde{ Y_t} = Y_t - \phi Y_{t-1}\)` and `\(\alpha_0 = (1 - \phi ) \beta_0\)` `$$\tilde Y_t = \alpha_0 + \beta_1 \tilde X_t+ v_t$$` - Generalized Least Squares (GLS) estimation - As `\(\phi\)` is unknown this equation is _infeasible_ - it cannot be directly estimated --- ## Estimation of dynamic causal effects - Strictly exogenous regressors 1. Estimate ADL representation - Simpler 2. GLS estimation (Cochrane-Orchutt) - More efficient - Depends on knowing the form of autocorrelation - GLS estimation consists of: 1. Transform model to get IID errors 1. estimate this model - Similar to weighted least squares and heteroscedasticity: - If form of heteroscedasticity is known - Estimate least squares, using a weighted sum of the residuals --- ## Iterated multiperiod forecasts - Assume AR(2) eq estimated: `$$Y_t = \beta_{0} +\beta_{1} Y_{t-1} +\beta_{2} Y_{t-2}+ u_{t}$$` - One period forecast using the last 2 period data `$$\hat Y_{T+1|T} = \hat \beta_{0} +\hat \beta_{1} Y_{T} + \hat \beta_{2} Y_{T-1}$$` - Use one period forecast for the next period: `$$\hat Y_{T+2|T} = \hat \beta_{0} + \hat \beta_{1} \hat Y_{T+1|T} +\hat \beta_{2} Y_{T}$$` - Use forecasts for the next period: `$$\hat Y_{T+3|T} = \hat \beta_{0} +\hat \beta_{1} \hat Y_{T+2|T} + \hat \beta_{2} \hat Y_{T+1|T}$$` --- ## Lag length selection - Bayesian Information criterion (or Schwartz) `$$\mathrm{BIC}(p) = \log\left[\frac{\mathrm{SSR}(p)}{T}\right]+\frac{\log(T)}{T}\left(p+1\right)$$` - Akaike Information criterion `$$\mathrm{AIC}(p) = \log\left[\frac{\mathrm{SSR}(p)}{T}\right]+\frac{2}{T}\left(p+1\right)$$` - Choose `\(p\)` to minimize criteria - First terms are decreasing and second terms are increasing in `\(p\)` - BIC gives fewer lags than AIC - `\(\mathrm{BIC}(p)\)` increases faster than `\(\mathrm{AIC}(p)\)` as `\(\log(T) > 2\)` --- ## Lag length selection in practice - Lag length selection with multiple predictors - In choosing ADL(p,q) model, select `\(p=q\)` that minimize information criterium --- ## VAR models - Forecasting model of a system with several variables - Variables are explained by lags of all variables - Only autoregressive - Each equation can be estimated by OLS - Used for macroeconomic forecasts - Has been used for other forecasts --- ## Vector Autoregressions (VAR) - VAR(1) model `$$Y_t = \beta_{10} +\beta_{11} Y_{t-1} + \gamma_{11} X_{t-1} + u_{1t}$$` `$$X_t = \beta_{20} +\beta_{21} Y_{t-1} + \gamma_{21} X_{t-1} + u_{2t}$$` - Simultaneous AR(1) - Coefficients can be estimated with OLS - Joint estimation of system to get joint distribution of all parameters - Adding a new variable `\(W_t\)` increases parameters - For VAR(1) it increases from 6 to 12! - For VAR(2) it increases from 10 to 21 - Use BIC for VAR to select number of lags