Question

In Example 10.4 , we saw that our estimates of the individual lag coefficients in a distributed lag model were very imprecise. One way to alleviate the multicollinearity problem is to assume that the \(\delta_{j}\) follow a relatively simple pattern. For concreteness, consider a model with four lags: $$y_{t}=\alpha_{0}+\delta_{0} z_{t}+\delta_{1} z_{t-1}+\delta_{2} z_{t-2}+\delta_{3} z_{t-3}+\delta_{4} z_{t-4}+u_{t}$$ Now, let us assume that the \(\delta_{j}\) follow a quadratic in the lag, \(j\) : $$ \delta_{j}=\gamma_{0}+\gamma_{1} j+\gamma_{2} j^{2} $$ for parameters \(\gamma_{0}, \gamma_{1},\) and \(\gamma_{2} .\) This is an example of a polynomial distributed lag (PDL) model. i. Plug the formula for each \(\delta_{j}\) into the distributed lag model and write the model in terms of the parameters \(\gamma_{h},\) for \(h=0,1,2\) ii. Explain the regression you would run to estimate the \(\gamma_{h}\) iii. The polynomial distributed lag model is a restricted version of the general model. How many restrictions are imposed? How would you test these? (Hint: Think \(F\) test.)

Short answer

Substitute the quadratic form of \(\delta_j\) into the lag model; estimate \(\gamma_h\) using linear regression on transformed variables; test three restrictions using an \(F\) test on the PDL model.

Step-by-step solution

Substitute \(\delta_j\)

Substitute the expression for \(\delta_{j}\), \(\delta_{j} = \gamma_{0} + \gamma_{1} j + \gamma_{2} j^{2}\), into the equation for each lag term in the model:\[\begin{align*}y_{t} &= \alpha_{0} + (\gamma_{0} + \gamma_{1}\cdot0 + \gamma_{2}\cdot0^2)z_{t} \&\quad + (\gamma_{0} + \gamma_{1}\cdot1 + \gamma_{2}\cdot1^2)z_{t-1} \&\quad + (\gamma_{0} + \gamma_{1}\cdot2 + \gamma_{2}\cdot2^2)z_{t-2} \&\quad + (\gamma_{0} + \gamma_{1}\cdot3 + \gamma_{2}\cdot3^2)z_{t-3} \&\quad + (\gamma_{0} + \gamma_{1}\cdot4 + \gamma_{2}\cdot4^2)z_{t-4} + u_{t}.\end{align*}\]

Simplify the Model

Simplify the expanded terms into an expression that is linear in \(\gamma_{0}, \gamma_{1},\) and \(\gamma_{2}\):\[\begin{align*}y_{t} &= \alpha_{0} + \gamma_{0}(z_{t} + z_{t-1} + z_{t-2} + z_{t-3} + z_{t-4}) \&\quad + \gamma_{1}(0z_{t} + 1z_{t-1} + 2z_{t-2} + 3z_{t-3} + 4z_{t-4}) \&\quad + \gamma_{2}(0^2z_{t} + 1^2z_{t-1} + 2^2z_{t-2} + 3^2z_{t-3} + 4^2z_{t-4}) + u_{t}.\end{align*}\]

Setup Regression Model

The model can be rewritten for regression as:\[y_{t} = \alpha_{0} + \gamma_{0}X_{0t} + \gamma_{1}X_{1t} + \gamma_{2}X_{2t} + u_{t}\]where \[\begin{align*}X_{0t} &= z_{t} + z_{t-1} + z_{t-2} + z_{t-3} + z_{t-4}, \X_{1t} &= 0z_{t} + 1z_{t-1} + 2z_{t-2} + 3z_{t-3} + 4z_{t-4}, \X_{2t} &= 0^2z_{t} + 1^2z_{t-1} + 2^2z_{t-2} + 3^2z_{t-3} + 4^2z_{t-4}.\end{align*}\]

Explain the Regression

To estimate the \(\gamma_{h}\) parameters, run a linear regression with \(y_{t}\) as the response variable and \(X_{0t}, X_{1t}, X_{2t}\) as the predictor variables. This regression will provide estimates for \(\alpha_{0}, \gamma_{0}, \gamma_{1},\) and \(\gamma_{2}\).

Identifying Restrictions

The PDL model imposes 3 restrictions on the underlying general model because the original model has 5 lag coefficients \(\delta_{j}\) but is reduced to 3 parameters \(\gamma_{0}, \gamma_{1}, \gamma_{2}\).

Test the Restrictions

To test these restrictions, perform an \(F\) test. The null hypothesis of the test is that the simple pattern does not hold, i.e., the PDL model is not a valid approximation of the unrestricted model. The alternative hypothesis is that the model is a valid simplification of the general model.

Key concepts

Multicollinearity

Multicollinearity is one of the main challenges you'll encounter when dealing with regression models. It occurs when independent variables in a model are highly correlated. This makes it difficult to determine the individual effect of each variable, as their influences may mirror or overlap with each other. In the context of a polynomial distributed lag (PDL) model, multicollinearity can lead to imprecise estimates of the lag coefficients, making it hard to make reliable inferences from the data.
To address multicollinearity, the PDL model simplifies the relationship among the lagged variables by assuming a specific pattern, such as a quadratic function as shown in the exercise. This reduces the number of parameters to estimate, which can provide more stable and interpretable results.

Lag Coefficients

Lag coefficients are parameters in a regression model that reflect the impact of past values of the independent variable on the current value of the dependent variable. In the polynomial distributed lag model, these lag coefficients are expressed as a polynomial function of the lag length. For example, if we assumed a quadratic form, each coefficient is defined as \[ \delta_j = \gamma_0 + \gamma_1 j + \gamma_2 j^2 \]where \( j \) is the lag number. This functional form simplifies the estimation process by transforming multiple potentially complex lag relationships into a manageable number of parameters. Using this approach allows for a more parsimonious model specification, helping to counteract issues like multicollinearity.

Regression Model

A regression model is a statistical tool used to explore the relationship between a dependent variable and one or more independent variables. In simple terms, it helps us understand how the dependent variable changes when any one of the independent variables is varied, holding the other independent variables constant.
In this particular case, the regression model is set up to estimate the parameters \( \gamma_0, \gamma_1, \gamma_2 \) within a PDL framework. The model takes the form:
\[ y_t = \alpha_0 + \gamma_0 X_{0t} + \gamma_1 X_{1t} + \gamma_2 X_{2t} + u_t \]
where \( X_{0t}, X_{1t}, X_{2t} \) are constructed from the lagged terms of the independent variable \( z_t \). This setup allows us to use linear regression techniques to find the best-fitting values of these parameters, making the complex problem of many lagged relationships more tractable.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide whether there is enough evidence to reject a null hypothesis. In the context of the PDL model, hypothesis testing can determine whether the restrictions imposed by the polynomial form are valid. Specifically, an \( F \) test can be employed to evaluate these restrictions.
The null hypothesis in this scenario is that the simplified PDL model is not significantly different from the unrestricted model, implying that it is not a suitable approximation. The alternative hypothesis is that the restrictions provide a good model fit, indicating that the simpler PDL model works as intended. By conducting an \( F \) test, researchers can assess the validity of these imposed restrictions and decide whether to accept or reject the simpler model form based on the resulting \( F \) statistic.