12 Worksheet 12: What’s new when applying linear regressions to time series data

12.1 Sections are under revision

12.2 Autoregressive distributed lag models

In previous worksheets, you have encountered autoregressive models. We are now extending the idea to cover the presence of regressors other than the lags of the regressand.

Autoregressive distributed lag models are used often in econometric practice. Use cases include determining whether there is additional information provided by other regressors in predicting the regressand (usually called testing for Granger causality), modeling dynamics involving two or more stationary time series (usually seen in macroeconometric models), or policy analysis (usually in cases where a regressor is some policy variable but you want to allow for effects that take time to manifest themselves).

The autoregressive distributed lag model most used in practice is usually called an ARDL(p,q), which has the form:

\[y_t=\alpha +\phi_1 y_{t-1}+\phi_2 y_{t-2}+\cdots \phi_p y_{t-p} +\delta_0 x_t+\delta_1 x_{t-1}+\cdots + \delta_q x_{t-q}+u_t\]

The term \(u_t\) is usually interpreted as some error term with certain statistical properties, in order for the coefficients to be meaningful. The ARDL(p,q) is actually relatively general and covers the following cases, which are usually found in the literature:

\(\phi_1=\phi_2=\cdots=\phi_p=0\): Finite distributed lag model
\(\phi_1=\phi_2=\cdots=\phi_p=0\) and \(\delta_1=\delta_2=\cdots=\delta_q=0\): Static model
\(\delta_1=\delta_2=\cdots=\delta_q=0\): Autoregressive model
\(\delta_0=0\): ARDL model used for predictive and forecasting purposes
Most general form can be used for policy analysis provided that \(x_t\) is strictly exogenous. This means that the error term satisfies \(\mathsf{E}\left(u_t|\mathsf{past\ history\ of\ y}, \mathsf{past\ history\ of\ x}, x_t, \mathsf{future\ values\ of\ x}\right)=0\).

OLS can be used to estimate coefficients provided that the assumptions for its use hold. The list would definitely include stationarity and weak dependence of both \(y\) and \(x\), no perfect collinearity among the \(x\)’s, and a suitable assumption about the error term. In predictive and forecasting settings, \(\mathsf{E}\left(u_t|\mathsf{past\ history\ of\ y}, \mathsf{past\ history\ of\ x}\right)=0\) would be enough.

In policy analysis settings, \(\mathsf{E}\left(u_t|\mathsf{past\ history\ of\ y}, \mathsf{past\ history\ of\ x}, x_t, \mathsf{future\ values\ of\ x}\right)=0\) is preferred. This ultimately boils down to the correct dynamic specification of the model, which is actually difficult to satisfy.

The choice of standard errors depends on assumptions about conditional heteroscedasticity and serial correlation. Dealing with conditional heteroscedasticity is already applied by default in cross-sectional settings (refer to vcovHC()). In time series settings and settings where financial data are used, it might be a good idea to be deliberate about thinking of a model for conditional heteroscedasticity. You have seen this already when you worked on the ACF of squared log returns.

In contrast, serial correlation is trickier to handle because its presence is usually an indication that the dynamic specification of the model may be incorrect. This can invalidate a policy analysis. In contrast, if there is serial correlation in a predictive or forecasting setting, all you need is to determine how much history for \(y\) and \(x\) you have to include in the model. If inference needs to be conducted in predictive or forecasting settings, then adjusting standard errors to account for the presence of serial correlation may restore valid inference (refer to vcovHAC()).

For a policy analysis, the presence of serial correlation may affect the interpretation of the “effects” and could invalidate inference. Therefore, if the goal is policy analysis, you need to spend time thinking about whether \(x\) is indeed strictly exogenous.

12.3 Exercise: What can we learn from these models?

Consider a finite distributed lag model of the form \(y_t=\delta_0 x_t+\delta_1 x_{t-1}+\delta_2 x_{t-2} +u_t\). Typically, \(y\) is what interests an analyst while \(x\) is something observable that can affect \(x\) or be used to predict \(y\). Note that \(u\) is unobservable that can affect \(y\) or is a prediction error.

In this exercise, you will consider a one unit temporary increase in \(u_t\), so that \(u_t^{\mathsf{new}}=u_t^{\mathsf{old}}+1\) but \(u_{t+k}^{\mathsf{new}}=u_{t+k}^{\mathsf{old}}\) for all \(k\geq 1\). Assume that everything else is held constant.

Explain why this change was framed as a temporary increase.
\(u_t\) is sometimes given a “shock” interpretation, meaning that there is some value of \(u_t\) but they do not know what value it could take and \(\mathsf{E}\left(u_t|\mathsf{past\ history\ of\ x}, x_t\right)=0\). Recalling the previous worksheet on optimal prediction, explain why \(u_t\) can be given a “shock” interpretation.
Using the given finite distributed lag model, write down expressions for the old and new values (after the one-unit temporary increase in \(u_t\)) of \(y_t\). Afterwards, show that \(y_t^{\mathsf{new}}-y_t^{\mathsf{old}}=1\) but \(y_{t+k}^{\mathsf{new}}-y_{t+k}^{\mathsf{old}}=0\) for all \(k\geq 1\).

Consider next a one unit permanent increase in \(u_t\). Assume that everything else is held constant.

Formulate in symbols what this would mean in terms of the present and future values of \(u_t\). Afterwards, show that \(y_{t+k}^{\mathsf{new}}-y_{t+k}^{\mathsf{old}}=1\) for all \(k\geq 0\).

12.4 Exercise: What can we learn from these models?

Continue from the previous exercise.

Consider next a one unit temporary increase in \(x_t\). Assume that everything else is held constant.

Formulate in symbols what this would mean in terms of the present and future values of \(x_t\). Compute \(y_{t+k}^{\mathsf{new}}-y_{t+k}^{\mathsf{old}}\) for all \(k\geq 0\).
After obtaining your responses to the previous item, find the immediate impact of a one unit temporary increase in \(x\) on \(y\), holding everything else constant. This is called the period-0 dynamic multiplier or impact multiplier. How are these tied to the parameters of the model?
Find the impact one and two periods later. These are called the period-1 and period-2 dynamic multipliers, respectively. How are these tied to the parameters of the model?
Find the impact three or more periods later. How are these tied to the parameters of the model?

12.5 Exercise: What can we learn from these models?

Continue from the previous exercise.

Consider next a one unit permanent increase in \(x_t\). Assume that everything else is held constant.

Formulate in symbols what this would mean in terms of the present and future values of \(x_t\). Compute \(y_{t+k}^{\mathsf{new}}-y_{t+k}^{\mathsf{old}}\) for all \(k\geq 0\).
After obtaining your responses to the previous item, find the immediate impact of a one unit permanent increase in \(x\) on \(y\), holding everything else constant. This is called the period-0 dynamic multiplier or impact multiplier. How are these tied to the parameters of the model?
Find the impact \(k\geq 1\) periods later, i.e. find all the period-\(k\) dynamic multipliers.
Find the cumulative multiplier or long-run multiplier or long-run propensity, which is the long-run impact of a one-unit permanent increase in \(x\) on \(y\). How are these tied to the parameters of the model?

12.6 Exercise: Alternative approaches for obtaining the long-run multiplier

Make sure you have completed the previous exercises before proceeding.

Consider a finite distributed lag model of the form \(y_t=\delta_0 x_t+\delta_1 x_{t-1}+\delta_2 x_{t-2} +u_t\).

Let \(y^*\) and \(x^*\) be the long-run or steady state values of \(y\) and \(x\). One way to think of the long-run or the steady state, there are no more “shocks”. This means that in the long -run, the \(u\)’s are equal to zero. In addition, the values of \(y\) and \(x\) are no longer changing and departing from their steady state values.

Using this alternative approach, start from the finite distributed lag model and obtain the long-run or steady state equation. After that, find \(\Delta y^*/\Delta x^*\) using this long-run equation.

12.7 Exercise: What regression should you run to get long-run multipliers?

There are two ways to obtain these long-run multipliers directly from a regression table.

What is the most straightforward way of computing a long-run multiplier from a regression table? Describe how from a table of regression results.
Define \(\theta=\delta_0+\delta_1+\delta_2\). Rewrite the finite distributed lag model into a form where \(\theta\) will appear. Consider substituting \(\delta_0=\theta-\delta_1-\delta_2\) into the finite distributed lag model.
Using the previous item, what kind of regression should you run to obtain the long-run multiplier directly? What new variables should you introduce?
Discuss the advantages of formulating the regression as found in the previous item. Are there interpretations for the coefficients of the other regressors?
Another version is to first show that the finite distributed lag model in this exercise could be written as \(y_t=\delta_0 \Delta x_t+\left(\delta_0+\delta_1\right)\Delta x_{t-1}+\left(\delta_0+\delta_1+\delta_2\right) x_{t-2}+u_t\). What regression should you run in this case in order to obtain the long-run multiplier directly? What are the new regressors here and what would be the interpretation for the coefficients of these regressors?