Chapter 16 – Distributed Lag Models

\chapter{Distributed lag models\label{lag.chap}} \section{Overview}
In many economic relationships, events occurring in the past have an impact on current outcomes. For example, the life cycle and permanent income hypotheses both suggest that current consumption spending will be affected by past income levels.\footnote{%
To be more precise, the permanent income hypothesis suggests that consumption spending is a function of the expected level of \textquotedblleft permanent income.\textquotedblright\ In practice, however, most analysts assume that the expected level of permanent income is a function of past levels of realized income. Thus, this relationship reduces to one in which past income levels are assumed to affect current consumption spending.} The accelerator model suggests that current investment spending is affected by previous levels of GDP. To take these factors into account, many time-series models include \textbf{lagged variables}. A lagged variable is denoted by a subscript that equals $t-k$ for some positive value of $k$.
A lagged relationship may be expressed as:
\begin{equation*}
Y_{t}=\beta _{o}+\beta _{1}X_{t}+\beta _{2}X_{t-1}+\beta _{3}Z_{t-3}+u_{t} \end{equation*}%
In this equation, it is assumed that the current value of $Y_{t}$ is affected by the current value of $X$, the value of $X$ occurring one-period in the past, and the value of $Z$ that occurred three time periods in the past.
Econometric models that include several lags of one or more independent variables are referred to as \textbf{distributed lag models}. The consumption function:
\begin{equation}
C_{t}=\alpha +\beta _{o}YD_{t}+\beta _{1}YD_{t-1}+\beta _{2}YD_{t-2}+\ldots +\beta _{k}YD_{t-k}+u_{t} \label{cons.dis.lag.8} \end{equation}%
is an example of a distributed lag model. In this model, consumption expenditures are assumed to a function of current and past levels of real disposable personal income. In a model of this sort, a change in the current period’s disposable personal income generates an effect on consumption in the present and the next $k$ periods. Thus, the long-run effect resulting from a change in disposable income is greater than the short-run effect. In particular, the individual coefficients $\beta _{i}$ captures the short-run effects of a change in the level of past disposable personal income on current consumption:
\begin{equation*}
\beta _{i}=\frac{\Delta C_{t}}{\Delta YD_{t-i}} \end{equation*}%
If there is a permanent change in the level of disposable personal income, however, the long-run effect on consumption can be measured by adding the short-run effects together (since the current level of consumption is affected by changes in the level of both current and past levels of disposable personal income). Defining $\Delta YD^{p}$ as a permanent change in the level of personal disposable income, the long-run impact of a change in disposable personal income can be measured as:\footnote{\textbf{Proof:} \par
Suppose that $YD_{t}$ is initially in a long-run state of equilibrium at: $% YD_{o}.$ Assuming that $YD_{t}$ has remained at this level for at least the past $k$ periods, the expected level of consumption will equal: \begin{equation*}
C=\alpha +\beta _{o}YD_{o}+\beta _{1}YD_{o}+\beta _{2}YD_{o}+\ldots +\beta _{k}YD_{o}
\end{equation*}%
or:
\begin{equation}
C^{o}=\alpha +(\beta _{0}+\beta _{1}+\cdots +\beta _{k})YD_{o} \tag{(1)} \end{equation}%
If the level of $YD$ rises to $YD_{1}$, and remains at this level for at least $k$ periods, then (after at least $k$ periods have elapsed) the new long-run level of consumption will be:
\begin{equation}
C^{1}=\alpha +(\beta _{0}+\beta _{1}+\cdots +\beta _{k})YD_{1} \tag{(2)} \end{equation}%
Thus, subtracting equation (2) from equation (1) results in: \begin{equation*}
C^{o}-C^{1}=(\beta _{0}+\beta _{1}+\cdots +\beta _{k})(YD_{o}-YD_{1}) \end{equation*}%
or:
\begin{equation*}
\frac{\Delta C}{\Delta YD^{p}}=\beta _{0}+\beta _{1}+\cdots +\beta _{k} \end{equation*}%
where $\Delta YD^{p}$ represents a permanent change in the level of disposable personal income.}
\begin{equation*}
\frac{\Delta C}{\Delta YD^{p}}=\beta _{o}+\beta _{1}+\cdots +\beta _{k} \end{equation*}
As long as all of the other assumptions of the classical regression model are satisfied, the parameters of distributed lag models (such as the one represented in equation \ref{cons.dis.lag.8}) may be estimated by OLS estimation techniques. There are, however, three problems that often occur when estimating distributed lag models:
\begin{itemize}
\item The use of lagged variables reduces the number of usable observations.
\item A multicollinearity problem often appears when a large number of lags are used.
\item An autocorrelation problem is often present in time series models of this sort.
\end{itemize}
Let’s discuss each of these issues.
Whenever lagged variables are used, the number of observations that can be included in the regression declines by $k$ observations (where $k$ is the length of the longest lag length for any independent variable). In particular, each additional lag results in a loss of one additional observation at the beginning of the sample. For example, suppose that you have 30 years of time series observations on a variable $X_{t}$ (for the period 1971-2000). Under these conditions, the variable $X_{t-1}$ has only 29 usable observations (since $X_{t-1}$ is not observed for the year 1971).
The variable $X_{t-3}$ will only have 27 observations. This relationship is depicted in Table \ref{us.obs.8}. If the variable $X_{t-3}$ is used as an independent variable in a regression, the years 1971-1973 cannot be included in the regression.
%TCIMACRO{%
%\TeXButton{tabular}{\begin {table}
%\begin{center}
%\begin{tabular}{|ccccc|} \hline
%\bf Year & \bf $X_t$ & \bf $X_{t-1}$ & \bf $X_{t-2}$ & \bf $X_{t-3}$ \\ \hline %1971 & 12.2 & – & – & – \\ %1972 & 11.8 & 12.2 & – & – \\ %1973 & 13.2 & 11.8 & 12.2 & – \\ %1974 & 14.5 & 13.2 & 11.8 & 12.2 \\ %$\vdots $ & $\vdots $ & $\vdots $ & $\vdots $ & \\ %1997 & 29.3 & 30.2 & 28.5 & 25.9 \\ %1998 & 32.3 & 29.3 & 30.2 & 28.5 \\ %1999 & 36.8 & 32.3 & 29.3 & 30.2 \\ %2000 & 37.2 & 36.8 & 32.3 & 29.3 \\ \hline %\end{tabular}
%\caption{Loss of observations when lagged variables are used \label{us.obs.8}} %\end{center}
%\end{table}}}%
%BeginExpansion
\begin {table}
\begin{center}
\begin{tabular}{|ccccc|} \hline
\bf Year & \bf $X_t$ & \bf $X_{t-1}$ & \bf $X_{t-2}$ & \bf $X_{t-3}$ \\ \hline 1971 & 12.2 & – & – & – \\ 1972 & 11.8 & 12.2 & – & – \\ 1973 & 13.2 & 11.8 & 12.2 & – \\ 1974 & 14.5 & 13.2 & 11.8 & 12.2 \\ $\vdots $ & $\vdots $ & $\vdots $ & $\vdots $ & \\ 1997 & 29.3 & 30.2 & 28.5 & 25.9 \\ 1998 & 32.3 & 29.3 & 30.2 & 28.5 \\ 1999 & 36.8 & 32.3 & 29.3 & 30.2 \\ 2000 & 37.2 & 36.8 & 32.3 & 29.3 \\ \hline \end{tabular}
\caption{Loss of observations when lagged variables are used \label{us.obs.8}} \end{center}
\end{table}%
%EndExpansion
Note that extending the length of the lag length by one period in a distributed lag model results in the loss of two degrees of freedom. One degree of freedom is lost as a result of a reduction in the size of the usable sample, the other is lost as a result of the additional parameter that must be estimated. This reduction in degrees of freedom can be quite significant when dealing with annual data with a limited number of observations.
A multicollinearity problem often appears when distributed lag models are estimated because the variables $X_t,$ $X_{t-1},\ldots ,$ $X_{t-k}$ are often highly correlated. The correlation among these variables is primarily the result of the substantial trend component that occurs in a large proportion of economic time series. When multicollinearity is present, parameter estimates become less precise.\footnote{% The effects of multicollinearity are discussed in more detail in Chapter \ref% {multicol.chap}.}
%TCIMACRO{%
%\TeXButton{cobwebmodel}{\exbox{The econometrics of cobwebs}{ %Markets for highly educated worker are often believed to be subject to a “lagged %supply” response. This means that a current increase in starting salaries will have %a larger effect on future supply than on current supply. For example, suppose %that a shortage appears in the market for engineers. In the short run, salaries may %increase quite dramatically, but few additional engineers will appear. Higher starting %salaries for engineers, however, will induce additional workers to major in %engineering. The quantity of engineers supplied, however, will not increase until %these students graduate from engineering school several years in the future.
%
%This lagged supply response indicates that the current supply of workers in affected %markets is determined, in part, by wages that were offered in past years. This provides %a natural application of lagged independent variables. In several studies, %Freeman (1971, 1972, 1975a, 1975b, 1976) provides an %interesting examination of how supply equations can be specified in such markets.
%Freeman finds evidence that a lagged supply response plays an %important role in explaining the supply of engineers, lawyers, physicists, psychologists, %and college graduates in general.
%
%Models of this type are referred to as “cobweb models” because the graph of the %adjustment process (using demand and supply diagrams) resembles a cobweb. Markets %such as this are often characterized by cycles in which periods of shortage and %increasing real wages are followed by shortages and declining real wages. (Cobweb %models are also used to explain the market for agricultural products.) %}}}%
%BeginExpansion
\exbox{The econometrics of cobwebs}{
Markets for highly educated worker are often believed to be subject to a “lagged supply” response. This means that a current increase in starting salaries will have a larger effect on future supply than on current supply. For example, suppose that a shortage appears in the market for engineers. In the short run, salaries may increase quite dramatically, but few additional engineers will appear. Higher starting salaries for engineers, however, will induce additional workers to major in engineering. The quantity of engineers supplied, however, will not increase until these students graduate from engineering school several years in the future.
This lagged supply response indicates that the current supply of workers in affected markets is determined, in part, by wages that were offered in past years. This provides a natural application of lagged independent variables. In several studies, Freeman (1971, 1972, 1975a, 1975b, 1976) provides an interesting examination of how supply equations can be specified in such markets.
Freeman finds evidence that a lagged supply response plays an important role in explaining the supply of engineers, lawyers, physicists, psychologists, and college graduates in general.
Models of this type are referred to as “cobweb models” because the graph of the adjustment process (using demand and supply diagrams) resembles a cobweb. Markets such as this are often characterized by cycles in which periods of shortage and increasing real wages are followed by shortages and declining real wages. (Cobweb models are also used to explain the market for agricultural products.) }%
%EndExpansion
If one is willing to make assumptions about the relationship that exists among the parameters in a distributed lag model then it is possible to transform the models into a form that requires fewer parameters. Two common transformations for this purpose are the Koyck and Almon lag models. These models are discussed below in sections \ref{Koyck.8} and \ref{Almon.8}.
A GLS estimation technique can be used to deal with the existence of autocorrelation (as discussed in Chapter \ref{auto.chap}).
\section{Example: Consumption function}
Let’s consider the distributed lag version of the consumption function in equation \ref{cons.dis.lag.8}. In particular, let’s assume that real consumption spending is a function of the current and the past five years’
real disposable personal income. When the parameters of this equation are estimated using data from 1929-2002, the resulting equation is:\footnote{% The data used to estimate this model (and the other consumption function estimates appearing below) is contained in the file \textquotedblleft gdp.dat\textquotedblright . A description of this variable appears in Table % \ref{gdp.dat} in Appendix \ref{data.appendix}. The consumption and disposable personal income variables are measured in billions of chained 1996 dollars.}
\begin{equation}
\hat{C}_{t}=\underset{(-3.02)}{-63.119}+\underset{(4.73)}{0.952}YD_{t}+% \underset{(0.05)}{0.0132}YD_{t-1}-\underset{(-0.33)}{0.099}YD_{t-2} \label{est.cons.8}
\end{equation}%
\begin{equation*}
-\underset{(-0.43)}{0.135}YD_{t-3}+\underset{(0.26)}{0.082}YD_{t-4}+\underset% {(0.55)}{0.116}YD_{t-5}
\end{equation*}%
\begin{equation*}
\text{(}t\text{-ratios in parentheses)}
\end{equation*}%
\begin{equation*}
\overline{\text{R}}^{2}=0.997
\end{equation*}%
\begin{equation*}
\text{Durbin-Watson statistic = 0.239}
\end{equation*}%
While observations are available for 74 years, the regression is based on a sample consisting of only 69 observations as a result of the five-year lag in the distributed lag model.
The only significant variable in equation \ref{est.cons.8} is the current year’s level of disposable income ($YD_{t}$). It is quite likely, however, that the low $t$-statistics for the lagged values of disposable income are the result of multicollinearity among the variables.\footnote{% The effect of multicollinearity on parameter estimates and standard errors is addressed in Chapter \ref{multicol.chap}. It is important to recall that OLS parameter estimates remain unbiased, consistent, and efficient even when multicollinearity is present (as long as there are no other violations of the assumptions of the classical regression model).
\par
To test for the presence of multicollinearity, an auxiliary regression of $% YD_{t}$ on the first five lagged values of itself was estimated. The $% \overline{\text{R}}^{2}$for this auxiliary regression is 0.999. As noted in Chapter \ref{multicol.chap}, such a high R$^{2}$ is strongly suggestive of a multicollinearity problem.} As noted above, multicollinearity is often found when distributed lag models are estimated. As discussed in The low Durbin-Watson statistic also indicates the existence of an autocorrelation problem.
\section{Caution: Defining the appropriate sample when lags are present} When estimating time series models, econometricians must be careful to properly define the sample that is used for estimation purposes. Most computer software packages allow users to create lagged variables using relatively simple commands. Some computer software packages automatically reduce the size of the sample by one for each time period that a variable is lagged. Many packages, however, fill the missing observations with zeros (or a numeric code indicating a missing value) when lagged variables are created. When this occurs, the user must be careful to appropriately redefine the sample before any estimation takes place. To determine how your statistical package deals with lagged variables, either check the manual that accompanies the package or examine the regression printout to see how many observations were used to generate the estimates. If the number of observations is not reduced by the maximum number of lags used in your model, then it is necessary for you to restrict the sample before estimating the parameters of models containing lagged variables..
In estimating the consumption function above (equation \ref{est.cons.8}), it was necessary to restrict the sample so that the first five observations were not included in the regression equation.
\section{Koyck lag\label{Koyck.8}}
As noted above, distributed lag models may be estimated directly by OLS\ methods. This estimation, however, suffers from a substantial loss of degrees of freedom when the length of the lag structure, $k$, is large.
Models of this sort are also likely to be subject to a multicollinearity problem for large values of $k$. The Koyck lag model is designed to deal with these problems by substantially reducing the number of model parameters.To illustrate this model, it will be helpful to reconsider the distributed lag version of the consumption function: \begin{equation}
C_{t}=\alpha +\beta _{o}YD_{t}+\beta _{1}YD_{t-1}+\beta _{2}YD_{t-2}+\ldots +\beta _{k}YD_{t-k}+u_{t} \label{cons.dis.lag.8a} \end{equation}
In most applications, events occurring in the recent past have a larger impact on current outcomes than events occurring in the distant past. Thus, in equation \ref{cons.dis.lag.8a} it is likely that the coefficients $\beta _i$ become smaller as $i$ increases. In other words: \begin{equation*}
\beta _o>\beta _1>\beta _2>\ldots >\beta _k \end{equation*}
The Koyck lag model is based on the assumption that the $\beta _i$ coefficients decline geometrically. Specifically, is assumed that each of the $\beta _i$ coefficients can be expressed as:\footnote{% This model is also called a “geometric lag model” since it is assumed that the distributed lag parameters decline geometrically.} \begin{equation}
\beta _i=\lambda ^i\beta _o \label{koyck.8a} \end{equation}
\begin{equation*}
\text{where }0<\lambda <1
\end{equation*}
Note that this assumption requires that $\beta _i$ declines as $i$ increases. A large value of $\lambda $ (close to one) indicates that the effect of past changes declines relatively slowly with time. A small value is consistent with a rapid decline in the effects resulting from changes in the independent variable.
Using the relationship in equation \ref{koyck.8a}, equation \ref% {cons.dis.lag.8a} can be rewritten as:
\begin{equation}
C_t=\alpha +\beta _oYD_t+\lambda \beta _oYD_{t-1}+\lambda ^2\beta _oYD_{t-2}+\ldots +\lambda ^k\beta _oYD_{t-k}+u_t \label{cons.dis.lag.8b} \end{equation}
Since this relationship is assumed to hold for all time periods, the relationship in period $t-1$ can be expressed as: \begin{equation}
C_{t-1}=\alpha +\beta _oYD_{t-1}+\lambda \beta _oYD_{t-2}+\lambda ^2\beta _oYD_{t-3} \label{cdl.8c}
\end{equation}
\begin{equation*}
+\ldots +\lambda ^k\beta _oYD_{t-(k+1)}+u_{t-1} \end{equation*}
To simplify the model, it is helpful to multiply both sides of equation \ref% {cdl.8c} by $-\lambda $ to form:
\begin{equation}
-\lambda C_{t-1}=-\lambda \alpha -\lambda \beta _oYD_{t-1}-\lambda ^2\beta _oYD_{t-2}-\lambda ^3\beta _oYD_{t-3} \label{cdl.8d} \end{equation}
\begin{equation*}
-\ldots -\lambda ^{k+1}\beta _oYD_{t-(k+1)}-\lambda u_{t-1} \end{equation*}
Adding equations \ref{cons.dis.lag.8b} and \ref{cdl.8d}, we have: \begin{equation*}
C_t-\lambda C_{t-1}=\alpha (1-\lambda )+\beta _oYD_t-\lambda ^{k+1}\beta _oYD_{t-(k+1)}+(u_t-\lambda u_{t-1})
\end{equation*}
Under the Koyck lag model, the term $-\lambda ^{k+1}$is close to zero for large values of $k$.\footnote{%
In some models, it is assumed that the distributed lag replationship has an infinite order (\textit{i.e.} $k=\infty $). In this case, $\lambda ^{k+1}$ will equal zero for $\left| \lambda \right| <1$.} Thus, to simplify the model, this term is set to zero, resulting in: \begin{equation}
C_t=\alpha (1-\lambda )+\lambda C_{t-1}+\beta _oYD_t+(u_t-\lambda u_{t-1}) \label{cdl.8e}
\end{equation}
By using the definitions:
\begin{equation*}
\gamma _o=\alpha (1-\lambda )
\end{equation*}
\begin{equation*}
\gamma _1=\lambda
\end{equation*}
\begin{equation*}
\gamma _2=\beta _o
\end{equation*}
\begin{equation*}
\epsilon _t=u_t-\lambda u_{t-1}
\end{equation*}
equation \ref{cdl.8e} may be stated as:
\begin{equation}
C_t=\gamma _o+\gamma _1C_{t-1}+\gamma _2YD_t+\epsilon _t \label{cdl.8f} \end{equation}
The parameters of equation \ref{cdl.8f} may be estimated by OLS\ techniques.% \footnote{%
Since successive error terms are correlated, however, one of the assumptions of the classical regression model are violated. The implications of this will be discussed in more detail below.} Using the definitions above, these estimates can be used to construct estimates of the original parameters: \begin{equation*}
\hat{\alpha}=\frac{\hat{\gamma}_o}{1-\hat{\gamma}_1} \end{equation*}
\begin{equation*}
\hat{\beta}_o=\hat{\gamma}_2
\end{equation*}
\begin{equation*}
\hat{\beta}_1=\hat{\gamma}_2\hat{\gamma}_1
\end{equation*}
\begin{equation*}
\hat{\beta}_2=\hat{\gamma}_2\hat{\gamma}_1^2 \end{equation*}
\begin{equation*}
\vdots
\end{equation*}
\begin{equation*}
\hat{\beta}_k=\hat{\gamma}_2\hat{\gamma}_1^k \end{equation*}
There are two major advantages of using the Koyck lag model: \begin{enumerate}
\item Under this model, only one lag is used in the estimated equation. This increases the number of observations that can be used to estimate model parameters.
\item The use of this procedure reduces the number of parameters that must be estimated. This reduces the potential for a multicollinearity problem and increases the degrees of freedom for the parameter estimates.
\end{enumerate}
There are, however, a number of serious problems associated with the Koyck lag procedure:
\begin{itemize}
\item This procedure requires a very strong assumption about the relationship among the parameters.
\item The residual in the estimated model (equation \ref{cdl.8f} in this example) must satisfy the assumptions of the classical regression model. In particular, the residual ($\epsilon _{t}$) must be independent of all independent variables. Since $\epsilon _{t}=u_{t}-\lambda u_{t-1}$, this residual is, in fact, expected to be correlated with the lagged dependent variable appearing on the right-hand side of equation \ref{cdl.8f} (since $% C_{t-1}$ is affected by the value of $u_{t-1}$). Due to the presence of a lagged dependent variable under the Koyck specification, the usual Durbin-Watson statistic cannot be used to test for the presence of first-order autocorrelation. (Alternative tests for the presence of autocorrelation in models with a lagged dependent variable are discussed in Chapter \ref{auto.chap}.)
\end{itemize}
\subsection{Example: Consumption function}
Let’s use the Koyck lag specification to estimate the distributed lag consumption function model. In this case, the consumption function becomes: \begin{equation*}
C_{t}=\gamma _{o}+\gamma _{1}C_{t-1}+\gamma _{2}YD_{t}+\epsilon _{t} \end{equation*}%
When the parameters of this equation are estimated, the resulting equation is:
\begin{equation}
\hat{C}_{t}=-\underset{(-0.98)}{9.514}+\underset{(14.36)}{0.846}C_{t-1}+% \underset{(3.19)}{0.167}YD_{t} \label{c.koyck.8} \end{equation}%
\begin{equation*}
\text{(}t\text{-ratios in parentheses)}
\end{equation*}%
The results in equation \ref{c.koyck.8} may be used to construct estimates of the original parameters in the distributed lag model. Table \ref{tKoyck.8} contains a list of these estimated parameters. Since $\hat{\beta}_{i}\,$is the estimated coefficient on $YD_{t-i}$, these results suggests, that a one-dollar increase in disposable personal income today will result in: \begin{itemize}
\item \$0.167 in additional consumption spending during the current year; \item \$0.141 in additional consumption spending next year; \item \$0.120 in additional consumption spending in two years; \item \$0.101 in additional consumption in three years; \item \$0.086 in additional consumption in four years; and \item \$0.072 in additional consumption in five years.
\end{itemize}
An examination of Table \ref{tKoyck.8} suggests that the effect of past income on consumption declines over time.
%TCIMACRO{%
%\TeXButton{tabular}{\begin {table}
%\begin{center}
%\begin{tabular}{|l|} \hline
%\bf {Parameter estimates} \\ \hline
%$\hat \beta _o =\hat \gamma _2=0.167$ \\ %$\hat \beta_1 =\hat \gamma _2\hat \gamma _1 = 0.167 \times 0.846=0..141$ \\ %$\hat \beta_2 =\hat \gamma _2\hat \gamma _1^2 = 0.167 \times 0.846^2 =0.120$ \\ %$\hat \beta_3 =\hat \gamma _2\hat \gamma _1^3 = 0.167 \times 0.846^3 =0.101$ \\ %$\hat \beta_4 =\hat \gamma _2\hat \gamma _1^4 = 0.167 \times 0.846^4 =0.086$ \\ %$\hat \beta_5 =\hat \gamma _2\hat \gamma _1^5 = 0.167 \times 0.846^5 =0.072$ \\ \hline %\end{tabular}
%\caption{Koyck lag estimates of consumption function parameters \label{tKoyck.8}} %\end{center}
%\end{table}}}%
%BeginExpansion
\begin {table}
\begin{center}
\begin{tabular}{|l|} \hline
\bf {Parameter estimates} \\ \hline
$\hat \beta _o =\hat \gamma _2=0.167$ \\ $\hat \beta_1 =\hat \gamma _2\hat \gamma _1 = 0.167 \times 0.846=0..141$ \\ $\hat \beta_2 =\hat \gamma _2\hat \gamma _1^2 = 0.167 \times 0.846^2 =0.120$ \\ $\hat \beta_3 =\hat \gamma _2\hat \gamma _1^3 = 0.167 \times 0.846^3 =0.101$ \\ $\hat \beta_4 =\hat \gamma _2\hat \gamma _1^4 = 0.167 \times 0.846^4 =0.086$ \\ $\hat \beta_5 =\hat \gamma _2\hat \gamma _1^5 = 0.167 \times 0.846^5 =0.072$ \\ \hline \end{tabular}
\caption{Koyck lag estimates of consumption function parameters \label{tKoyck.8}} \end{center}
\end{table}%
%EndExpansion
Under the Koyck lag model, the long-run change in consumption resulting from a permanent change in the level of disposable income is simply the sum of the short-run effects. Using the analysis above, this can be computed as: \begin{equation*}
\frac{\Delta C}{\Delta YD^{P}}=\beta _{o}+\beta _{1}+\beta _{2}+\cdots +\beta _{k}
\end{equation*}%
\begin{equation*}
\text{where: }\Delta YD^{P}\text{ represents a permanent change in the level of disposable income}
\end{equation*}%
\begin{equation*}
\frac{\Delta C}{\Delta YD^{P}}=\beta _{o}(1+\lambda ^{2}+\lambda ^{3}+\cdots +\lambda ^{k})
\end{equation*}%
As long as $\lambda <1$ (and assuming that $k$ is large), this simplifies to:% \footnote{%
This can be demonstrated fairly simply. Define $S$ as: \begin{equation}
S=1+\lambda +\lambda ^{2}+\lambda ^{3}+\cdots +\lambda ^{k} \tag{(1)} \end{equation}%
Then,
\begin{equation}
-\lambda S=-\lambda -\lambda ^{2}-\lambda ^{3}-\lambda ^{4}-\cdots -\lambda ^{k+1} \tag{(2)}
\end{equation}%
Adding equations (1) and (2), results in:
\begin{equation*}
S-\lambda S=1-\lambda ^{k+1}
\end{equation*}%
If $k$ is large $\lambda <1$, then $\lambda ^{k+1}\approx 0$. Thus, \begin{equation*}
S-\lambda S\approx 1-\lambda ^{k+1}
\end{equation*}%
\par
Solving for $S$:
\begin{equation*}
S\approx \frac{1}{1-\lambda }
\end{equation*}%
Using the definition above,
\begin{equation*}
1+\lambda +\lambda ^{2}+\lambda ^{3}+\cdots +\lambda ^{k}\approx \frac{1}{% 1-\lambda }
\end{equation*}%
Thus,
\begin{equation*}
\beta _{o}\left( 1+\lambda +\lambda ^{2}+\lambda ^{3}+\cdots +\lambda ^{k}\right) \approx \frac{\beta _{o}}{1-\lambda } \end{equation*}%
}
\begin{equation*}
\frac{\Delta C}{\Delta YD}\approx \frac{\beta _{o}}{1-\lambda } \end{equation*}%
In the example above, the long-run effect resulting from an additional dollar of disposable income is:
%TCIMACRO{%
%\TeXButton{Cigarette ad ban}{\exbox{The Ban on Radio and TV Cigarette Advertising}{ %In response to growing public awareness of the health hazards associated with %the use of tobacco, radio and TV cigarette advertising was banned in 1970. It %was generally believed that the level of cigarette consumption would decline in %response to this ban.
%
%In a classic study, Hamilton (1972) provides evidence indicating that this ban %did not have its intended effect.
%Prior to the enactment of the ban, radio and television %stations were required to provide “balanced” coverage of all controversial %issues under the Fairness Doctrine enforced by the Federal Communications %Commission. In practice, this meant that radio and television stations were %required to provide free antismoking advertisements. Hamilton, using a distributed %lag model, finds that radio and TV advertising by cigarette producers had a relatively %small positive impact on sales. The “health scare” associated with the release of %medical studies, the Surgeon General’s report, and the advent of antismoking %advertising had a substantial adverse effect on sales.
%
%Since the ban on cigarette advertising also resulted in the elimination of the %free antismoking commercials, Hamilton’s study suggests that it actually %increased the sales (and profitability) associated with cigarette production.
%
%}} }%
%BeginExpansion
\exbox{The Ban on Radio and TV Cigarette Advertising}{ In response to growing public awareness of the health hazards associated with the use of tobacco, radio and TV cigarette advertising was banned in 1970. It was generally believed that the level of cigarette consumption would decline in response to this ban.
In a classic study, Hamilton (1972) provides evidence indicating that this ban did not have its intended effect.
Prior to the enactment of the ban, radio and television stations were required to provide “balanced” coverage of all controversial issues under the Fairness Doctrine enforced by the Federal Communications Commission. In practice, this meant that radio and television stations were required to provide free antismoking advertisements. Hamilton, using a distributed lag model, finds that radio and TV advertising by cigarette producers had a relatively small positive impact on sales. The “health scare” associated with the release of medical studies, the Surgeon General’s report, and the advent of antismoking advertising had a substantial adverse effect on sales.
Since the ban on cigarette advertising also resulted in the elimination of the free antismoking commercials, Hamilton’s study suggests that it actually increased the sales (and profitability) associated with cigarette production.
}
%EndExpansion
\begin{equation*}
\frac{\Delta C}{\Delta YD}\approx \frac{0.167}{1-0.846.}=1.08 \end{equation*}%
This result is somewhat troubling since an MPC\ \TEXTsymbol{>}\ 1 is inconsistent with economic theory. It suggests that the receipt by households of an additional dollar of disposable income results in an additional \$1.08 in consumption spending in the long run.
\section{Almon lag\label{Almon.8}}
The Almon lag model provides an alternative, and somewhat more flexible, procedure for estimating distributed lag models.\footnote{% This model was originally developed by Almon (1965). A good discussion appears in Kelejian and Oates (1989), pp. 170-178.} Once again, let’s consider the consumption function discussed above: \begin{equation}
C_{t}=\alpha +\beta _{o}YD_{t}+\beta _{1}YD_{t-1}+\beta _{2}YD_{t-2}+\ldots +\beta _{k}YD_{t-k}+u_{t} \label{pdl.8}
\end{equation}%
As under the Koyck lag model, the Almon lag model assumes that the parameters $\beta _{o},\beta _{1},\ldots ,\beta _{k}$ can be described by a simple function. Under the Almon lag procedure, it is assumed that the parameters satisfy one of the following polynomial relationships: \begin{equation*}
\begin{array}{ll}
\text{1st-order polynomial (linear)} & \beta _{i}=\gamma _{o}+\gamma _{1}i% \text{ (for }i=1,\ldots ,k)\text{ } \\
\text{2nd-order polynomial (quadratic)} & \beta _{i}=\gamma _{o}+\gamma _{1}i+\gamma _{2}i^{2} \\
\text{3rd-order polynomial (cubic)} & \beta _{i}=\gamma _{o}+\gamma _{1}i+\gamma _{2}i^{2}+\gamma _{3}i^{3} \\
\vdots & \vdots \\
m\text{th order polynomial(}m<k\text{)} & \beta _{i}=\gamma _{o}+\gamma _{1}i+\gamma _{2}i^{2}+\ldots +\gamma _{m}i^{m}% \end{array}%
\end{equation*}%
For this reason, the Almon lag structure is also commonly referred to as a \textbf{polynomial distributed lag (or PDL)}. When using this model, the econometrician estimates the parameters $\gamma _{1},\gamma _{2},\ldots ,\gamma _{m}$ and uses these estimates to generate estimates of the parameters $\beta _{o},\beta _{1},\ldots ,\beta _{k}$.
Figure~\ref{pdl_g_ots} illustrates possible linear, quadratic, and cubic polynomial lag relationships. The vertical axis on each of these graphs represents the distributed lag coefficients corresponding to the lag lengths appearing on the horizontal axis. Higher order polynomials provide better approximations to the underlying lag structure, but require the estimation of more parameters. Thus, higher order PDL models are more likely to be subject to the multicollinearity of limited degrees of freedom problem that often characterizes direct OLS\ estimation of the distributed lag parameters. In practice, econometricians tend to favor the use of lower order polynomials.
\begin{center}
\FRAME{ftbpF}{1.9009in}{4.7271in}{0pt}{}{\Qlb{pdl_g_ots}}{fig16-1.gif}{% \special{language “Scientific Word”;type “GRAPHIC”;maintain-aspect-ratio TRUE;display “USEDEF”;valid_file “F”;width 1.9009in;height 4.7271in;depth 0pt;original-width 3.6876in;original-height 9.25in;cropleft “0”;croptop “1”;cropright “1”;cropbottom “0”;filename
‘GRAPHS/Fig16-1.gif’;file-properties “XNPEU”;}} \end{center}
Under the Almon lag model, it is only necessary to estimate $m+1$ parameters ($\gamma _{o},\gamma _{1},\ldots ,\gamma _{m}$). The model simplifies the estimation process as long as the order of the polynomial ($m$) is less than the longest lag in the distributed lag model ($k$).\footnote{% If $m=k$ there would be no benefits to using this model. If $m>k$ the $% \gamma _{i}$ parameters could not be estimated (since there would be more parameters to estimate than there are variables). In this case, the model parameters are not identified.} Once the $m$th-order polynomial is estimated, it may be used to generate estimates of the $k+1$ parameters $% \beta _{o},\beta _{1},\ldots ,\beta _{k}$. Table \ref{poly.tab.8} illustrates the relationship between the parameters $\gamma _{o},\gamma _{1},\ldots ,\gamma _{m}$ and the underlying coefficients $\beta _{o},\beta _{1},\ldots ,\beta _{k}$ under the linear and quadratic models.
%TCIMACRO{%
%\TeXButton{tabular}{\begin {table}
%\begin{center}
%\begin{tabular}{|ccc|} \hline
%\bf Coefficient & \bf{1st-order PDL} & \bf {2nd-order PDL} \\ \hline %$\beta_o$ & $\gamma_o$ & $\gamma _o$ \\
%$\beta_1 $ & $\gamma_o+\gamma_1$ & $\gamma _o+\gamma _1+\gamma _2$ \\ %$\beta_2 $ & $\gamma_o+2\gamma_1$ & $\gamma _o+2\gamma _1+2^2\gamma _2$ \\ %$\vdots $ & $\vdots $ & $\vdots $ \\
%$\beta_k $ & $\gamma_o+k\gamma_1$ & $\gamma _o+k\gamma _1+k^2 \gamma _2$ \\ \hline %\end{tabular}
%\caption{Almon lag model – 1st order polynomial \label{poly.tab.8}} %\end{center}
%\end{table}}}%
%BeginExpansion
\begin {table}
\begin{center}
\begin{tabular}{|ccc|} \hline
\bf Coefficient & \bf{1st-order PDL} & \bf {2nd-order PDL} \\ \hline $\beta_o$ & $\gamma_o$ & $\gamma _o$ \\
$\beta_1 $ & $\gamma_o+\gamma_1$ & $\gamma _o+\gamma _1+\gamma _2$ \\ $\beta_2 $ & $\gamma_o+2\gamma_1$ & $\gamma _o+2\gamma _1+2^2\gamma _2$ \\ $\vdots $ & $\vdots $ & $\vdots $ \\
$\beta_k $ & $\gamma_o+k\gamma_1$ & $\gamma _o+k\gamma _1+k^2 \gamma _2$ \\ \hline \end{tabular}
\caption{Almon lag model – 1st order polynomial \label{poly.tab.8}} \end{center}
\end{table}%
%EndExpansion
Let’s examine the process of estimating the parameters of an Almon lag model.
\subsubsection{Example: Consumption function} Equation \ref{est.cons.8} contained an estimated distributed lag consumption function model. In this model, the level of current real consumption expenditures was assumed to be affected by the current and past five years levels of real disposable income. This model may be expressed as: \begin{equation}
C_{t}=\alpha +\beta _{o}YD_{t}+\beta _{1}YD_{t-1}+\beta _{2}YD_{t-2}+\ldots +\beta _{5}YD_{t-5}+u_{t} \label{cons.almon.8} \end{equation}%
Suppose that we wished to approximate the parameters of this model using a second-order polynomial distributed lag. Using the definitions in Table \ref% {poly.tab.8}, equation \ref{cons.almon.8} can be restated as: \begin{equation}
C_{t}=\alpha +\gamma _{o}YD_{t}+\ \left( \gamma _{o}+\gamma _{1}+\gamma _{2}\right) YD_{t-1} \label{c.a.8a}
\end{equation}%
\begin{equation*}
+\left( \gamma _{o}+2\gamma _{1}+2^{2}\gamma _{2}\right) YD_{t-2}+\ldots \end{equation*}%
\begin{equation*}
+\left( \gamma _{o}+5\gamma _{1}+5^{2}\gamma _{2}\right) YD_{t-5}+u_{t} \end{equation*}%
With a little bit of algebraic manipulation, equation \ref{c.a.8a} may be rewritten as:
\begin{equation}
C_{t}=\alpha +\gamma _{o}\sum_{i=0}^{5}YD_{t-i}+\gamma _{1}\sum_{i=1}^{5}iYD_{t-i}+\gamma _{2}\sum_{i=1}^{5}i^{2}YD_{t-i}+u_{t} \label{c.a.8b}
\end{equation}%
By defining new variables $Z_{ot},Z_{1t},$ and $Z_{2t}$ as: \begin{equation*}
Z_{ot}=\sum_{i=0}^{5}YD_{t-i}
\end{equation*}%
\begin{equation*}
Z_{1t}=\sum_{i=1}^{5}iYD_{t-i}
\end{equation*}%
and
\begin{equation*}
Z_{2t}=\sum_{i=1}^{5}i^{2}YD_{t-i}
\end{equation*}%
equation \ref{c.a.8b} may be simplified to: \begin{equation}
C_{t}=\alpha +\gamma _{o}Z_{ot}+\gamma _{1}Z_{1t}+\gamma _{2}Z_{2t}+u_{t} \label{poly.8ab}
\end{equation}%
The parameters of this equation may now be estimated by OLS techniques.
Since this model involves fewer estimated parameters than the original equation, the multicollinearity problem is reduced (and degrees of freedom are increased). The estimated parameters $\hat{\gamma}_{o},\hat{\gamma}_{1}$% , and $\hat{\gamma}_{2}$ may be used to formulate estimates of the underlying distributed lag relationship (using the formulas in Table \ref% {poly.tab.8}).
When the parameters of equation \ref{poly.8ab} are estimated, the resulting equation is:
\begin{equation}
\hat{C}_{t}=\underset{(-2.97)}{-60.7224}+\underset{(7.14)}{0.782466}Z_{o}-% \underset{(-4.55)}{0.651916}Z_{1}+\ \underset{(3.89)}{0.109413}Z_{2} \label{est.almon.8a}
\end{equation}%
\begin{equation*}
\overline{\text{R}}^{2}=0.997
\end{equation*}%
\begin{equation*}
\text{Durbin-Watson statistic = 0.220}
\end{equation*}%
In Table \ref{talmon.8} the results from equation \ref{est.almon.8a} are used to generate estimated values of the original model parameters. Once again, these results indicate that the effect of past income declines rather rapidly as the length of the lag increases. As noted above, the long-run effect on consumption of a permanent increase in the level of disposable income is given by the sum of the coefficients on lagged disposable income.
In this case, this long-run effect sum can be estimated as: %TCIMACRO{%
%\TeXButton{tabular}{\begin {table}
%\begin{center}
%\begin{tabular}{|l|} \hline
%\bf {Parameter estimates} \\ \hline
%$\hat \beta_o =\hat \gamma _o=0.782 $ \\
%$\hat \beta_1 =\hat \gamma _o+ \hat \gamma _1+\hat \gamma _2=0.782466-0.651916+0.068508=0.240 $ \\ %$\hat \beta_2 =\hat \gamma _o+2\hat \gamma _1+2^2\hat \gamma _2=0.782466-2(0.651916)+4(0.109413)= -0.084 $ \\ %$\hat \beta_3 =\hat \gamma _o+3\hat \gamma _1+3^2 \hat \gamma _2=0.782466-3(0.651916)+9(0.109413)= -0.189 $ \\ %$\hat \beta_4 =\hat \gamma _o+4\hat \gamma _1+4^2 \hat \gamma _2=0.782466-4(0.651916)+16(0.109413)= -0.075 $ \\ %$\hat \beta_5 =\hat \gamma _o+5\hat \gamma _1+5^2 \hat \gamma _2=0.782466-5(0.651916)+25(0.109413)= 0.258 $ \\ \hline %\end{tabular}
%\caption{Almon lag estimates of consumption function parameters \label{talmon.8}} %\end{center}
%\end{table}} }%
%BeginExpansion
\begin {table}
\begin{center}
\begin{tabular}{|l|} \hline
\bf {Parameter estimates} \\ \hline
$\hat \beta_o =\hat \gamma _o=0.782 $ \\
$\hat \beta_1 =\hat \gamma _o+ \hat \gamma _1+\hat \gamma _2=0.782466-0.651916+0.068508=0.240 $ \\ $\hat \beta_2 =\hat \gamma _o+2\hat \gamma _1+2^2\hat \gamma _2=0.782466-2(0.651916)+4(0.109413)= -0.084 $ \\ $\hat \beta_3 =\hat \gamma _o+3\hat \gamma _1+3^2 \hat \gamma _2=0.782466-3(0.651916)+9(0.109413)= -0.189 $ \\ $\hat \beta_4 =\hat \gamma _o+4\hat \gamma _1+4^2 \hat \gamma _2=0.782466-4(0.651916)+16(0.109413)= -0.075 $ \\ $\hat \beta_5 =\hat \gamma _o+5\hat \gamma _1+5^2 \hat \gamma _2=0.782466-5(0.651916)+25(0.109413)= 0.258 $ \\ \hline \end{tabular}
\caption{Almon lag estimates of consumption function parameters \label{talmon.8}} \end{center}
\end{table}
%EndExpansion
\begin{equation*}
0.782+0.240-0.084-0.189-0.075+0.258=0.932
\end{equation*}%
Under the original unrestricted distributed lag model appearing in equation % \ref{est.cons.8}, the estimated long-run effect can be determined directly by adding the estimated coefficients on the current and lagged values of $% YD_{t}$. The unrestricted model suggests a long-run effect equal to 0.929.
Thus, the estimated long-run effect resulting from a \$1 increase in disposable income is quite similar under the Almon lag model and the unrestricted model.
In summary, the estimation of an Almon lag model is conducted in the following manner:
\begin{enumerate}
\item[Step 1:] Determine the order of the polynomial used to approximate the distributed lag parameters. A higher-order polynomial will provide a more accurate approximation, but will be more susceptible to a multicollinearity problem (and will have fewer degrees of freedom).
\item[Step 2:] Replace the parameters of the original equation with the values defined by the polynomial relationship. Simplify the equation so that it is linear in these parameters. This will involve the creation of new variables $Z_o,Z_1,\ldots ,Z_m$ when an $m$th order polynomial is used.
\item[Step 3:] Estimate the parameters of the transformed model using an OLS\ estimation procedure.
\item[Step 4:] Use these estimates to generate the estimates of the original model parameters.
\end{enumerate}
In the past, the estimation of Almon lag models required a fair amount of work in creating the transformed variables and generating parameter estimates. Fortunately, several econometrics packages automatically generate estimates of polynomial lag models through the use of quite simple commands.% \footnote{%
EViews and LIMDEP, in particular, have particularly simple commands for estimating models of this sort. All transformations are done by these programs automatically.}
Caution:\
\section{Summary}
The use of distributed lag models make it possible to estimate relationships in which effects occur gradually over two or more periods of time. Since economic interactions take place over time, changes in many economic variables will have effects that will persist beyond the current time period. Thus, the use of distributed lag models makes it possible for econometric models to more accurately reflect real-world economic processes.
In practice, however, multicollinearity problems often make it difficult to provide reliable estimates of individual coefficients in distributed lag models. In such cases, econometricians have sometimes relied on the Koyck and Almon lag models to simplify the estimation process.
\section{Key Concepts}
distributed lag model
Koyck lag
Almon lag
polynomial distributed lag
\newpage\
\section{Exercises and problems}
\begin{enumerate}
\item Suppose that an econometrician estimates the following consumption function:
\begin{equation}
C_t=10.0+0.5YD_t+0.2\beta _1YD_{t-1}+0.1\beta _2YD_{t-2} \label{cons.forecast}
\end{equation}
\begin{equation*}
(\text{all variables are measured in billions}) \end{equation*}
\begin{enumerate}
\item Suppose that the level of $YD_t$ has been constant at $YD_t=100$ for each of the past 5 years and remains at that level today. Determine the level of consumption expenditures that is predicted by this estimated consumption function.
\item Suppose that disposable income rises to 110 during the current year and returns to 100 next year. Use the estimated consumption function in equation \ref{cons.forecast} to forecast the effect on consumption during the current and next 3 years.
\item Suppose instead, that a permanent change in the level of disposable personal income occurs from 100 to 110 during the current (and subsequent) years. Forecast the effect of this change on consumption in the current and next three years (using equation \ref{cons.forecast}).
\end{enumerate}
\item In a classic study, Andersen and Jordan (1968) attempt to measure the relative effectiveness of fiscal and monetary policy. One of the equations estimated in this study is:
\begin{equation*}
\widehat{\Delta \text{GNP}}_{t}=\underset{(2.16)}{1.99}+\underset{(2.17)}{% 1.57}\Delta \text{M}_{t}+\underset{(3.60)}{1.94}\Delta \text{M}_{t-1}+% \underset{(3.37)}{1.80}\Delta \text{M}_{t-2}+\underset{(1.88)}{1.28}\Delta \text{M}_{t-3}
\end{equation*}%
\begin{equation*}
-\underset{(0.65)}{0.15}\Delta (\text{R}_{t}-\text{E}_{t})-\underset{(1.08)}{% 0.20}\Delta \left( \text{R}_{t-1}-\text{E}_{t-1}\right) \end{equation*}%
\begin{equation*}
+\underset{(0.55)}{0.10}\Delta (\text{R}_{t-2}-\text{E}_{t-2})+\underset{% (0.45)}{0.47}\Delta (\text{R}_{t-3}-\text{E}_{t-3}) \end{equation*}%
\begin{equation*}
\text{(}t\text{-statistics in parentheses)} \end{equation*}%
\begin{equation*}
\begin{array}{ll}
\text{where:} & \widehat{\Delta \text{GNP}}_{t}\text{ = predicted quarterly change in nominal gross national product} \\ & \Delta \text{M}_{t}\text{ = quarterly change in the money stock } \\ & \text{R}_{t}\text{ = high employment government revenue in quarter }t \\ & \text{E}_{t}\text{ = high employment government expenditures in quarter }t \\
& \Delta (\text{R}_{t}-\text{E}_{t})\text{ = quarterly change in high employment government surplus}%
\end{array}%
\end{equation*}%
The \textquotedblleft high-employment\textquotedblright\ government revenue and expenditure variables are estimates of what government revenue and spending would be if the economy were at \textquotedblleft full employment.\textquotedblright\ The data used for this study consisted of quarterly data from the first quarter of 1952 to the second quarter of 1968.
\begin{enumerate}
\item What does economic theory predict about the signs of the coefficient in the equation above?
\item Why is a lagged relationship used?
\item Which coefficients are significant at a .05 significance level?
\item What do these results suggest about the relative importance of fiscal and monetary policy?

\end{enumerate}
\item A Koyck lag model of the consumption function is estimated as: \begin{equation*}
\hat{C}_{t}=10.23+0.62C_{t-1}+.92\text{\textit{YD}}_{t} \end{equation*}%
Use these estimates to derive the coefficients on the first four lags of \textit{YD}$.$
\item An analyst formulates a distributed lag consumption function model in which real consumption expenditures are assumed to be a function of the current and past 6 year’s levels of real disposable income. A second-order Almon lag model of the consumption function is estimated as: \begin{equation*}
C_{t}=5.23+0.82Z_{ot}+0.23Z_{1t}+0.02Z_{2t} \end{equation*}%
Use these results to construct estimates of the underlying distributed lag parameters.
\item
\begin{enumerate}
\item Use the \textquotedblleft money2.dat\textquotedblright\ file (described in Table \ref{money2.dat} in Appendix \pageref{data.appendix}) to estimate the parameters of the following model: \begin{equation*}
\text{GDP}_{t}\text{ = }\beta _{o}+\gamma _{o}\text{M}_{t}+\gamma _{1}\text{M% }_{t-1}+\ldots +\gamma _{8}\text{M}_{t-8}+u_{t} \end{equation*}
\item Examine the possibility of multicollinearity by computing an auxiliary regression in which M$_{t}$ is the dependent variable. Is there evidence of multicollinearity?
\item Reestimate this model using a 2nd order polynomial distributed lag model. Are the estimated parameters similar to those found in (a)?
\end{enumerate}
\item
\begin{enumerate}
\item Use the data in Table \ref{money2.dat} in Appendix \ref{data.appendix} (or the file \textquotedblleft money2.dat\textquotedblright ) to estimate the parameters of the following model:
\begin{equation*}
\text{GDP}_{t}\text{ = }\beta _{o}+\gamma _{o}\text{M}_{t}+\gamma _{1}\text{M% }_{t-1}+\ldots +\gamma _{8}\text{M}_{t-8}+u_{t} \end{equation*}
\item Reestimate this model using a Koyck lag model. Are the estimated parameters similar to those found in (a)?
\end{enumerate}
\item
\begin{enumerate}
\item Use the data in Table \ref{imports.dat} in Appendix \ref{data.appendix} (or the file \textquotedblleft imports.dat\textquotedblright ) to estimate the parameters of the following model:
\begin{equation*}
\text{Imports}_{t}\text{ = }\beta _{o}+\gamma _{o}\text{\textit{YD}}% _{t}+\gamma _{1}\text{\textit{YD}}_{t-1}+\ldots +\gamma _{5}\text{\textit{YD}% }_{t-5}+u_{t}
\end{equation*}
\item What rationale might be used to justify this specification?
\item Reestimate this model using a Koyck lag model. Are the estimated parameters similar to those found in (a)?
\item Reestimate this model using a 2nd-order polynomial distributed lag model. Compare your estimates with those in parts (a) and (b).
\end{enumerate}
\item Use the data in the \textquotedblleft cons.dat\textquotedblright\ file (described in Table \ref{cons.dat.app} in Appendix \ref{data.appendix} ) to verify the estimates appearing in equation \ref{c.koyck.8}.
\begin{enumerate}
\item Test for the presence of first-order autocorrelation at a 5\% significance level using one of the tests described in Chapter \ref% {auto.chap}. (Note: the Durbin-Watson statistic is inappropriate due to the presence of a lagged dependent variable.) \item If autocorrelation is found, apply an appropriate correction technique (as discussed in Chapter \ref{auto.chap}). Do the estimated coefficients change substantially? (Answer this part only if a correction is needed.) \end{enumerate}
\item Use the data in the \textquotedblleft cons.dat\textquotedblright\ file (described in Table \ref{cons.dat.app} in Appendix \ref{data.appendix}) and a polynomial distributed lag model to verify the estimates appearing in equation \ref{est.almon.8a}.
\item
\begin{enumerate}
\item Use the data in the \textquotedblleft cons.dat\textquotedblright\ file (described in Table \ref{cons.dat.app} in Appendix \ref{data.appendix}) to estimate a third-order polynomial distributed lag model for the consumption function appearing in equation \ref{cons.almon.8}.
\item Compare the estimated results from part (a) to those appearing in equation \ref{est.almon.8a}. Is a 3rd-order polynomial distributed lag model preferable to a 2nd-order polynomial distributed lag model in this case?
Explain.
\end{enumerate}
\end{enumerate}