19.2 Finite Distributed Lag Models

Distributed-lag models include past or lagged independent variables: \[y_t=\alpha+\beta_0 \cdot x_t+\beta_1 \cdot x_{t-1}+\beta_2 \cdot x_{t-2}+\dots \beta_k \cdot x_{t-k}+\epsilon\] There are many reasons to include lagged independent variables such as psychological (e.g., it is difficult to break a habit or adjust to a new situation), economic (e.g., contractual obligations), or political (e.g., effectiveness of policy builds up over time) reasons.

The relationship between income and consumption is used to introduce distributed lag models. Assume the following relationship between income and consumption: \[C_t=\alpha+\beta_0 \cdot I_t+\beta_1 \cdot I_{t-1}+\beta_2 \cdot I_{t-2}\] Assume that $\alpha_0=100$, $\beta_0=0.4$, $\beta_1=0.3$, and $\beta_2=0.2$. For this example, the following questions are of interest:

What is the long-run consumption with $4,000?
How does the consumption change if the income increases to $5000?

Note that the sum of the $\beta_i$’s is 0.9. The long-run multiplier (or long-run propensity) is written as: \[\sum_{i=1}^k \beta_i = \beta_0+\beta_1+\beta_2 + \dots +\beta_k = \beta\]

The question about how may lagged independent variables to include is a difficult to answer question. If the assumption is made that all $\beta_k$ are of the same sign, then the so-called Koyck transformation can be applied: \[\beta_k = \beta_0 \cdot \lambda^k \quad \text{for} \quad k=0,1,2,\dots\] Characteristics of this assumption:

$\lambda <1$ gives less weight to distant values of $\beta$
Long-run multiplier is finite, i.e., \[\sum_{k=0}^\infty \beta_k = \beta_0 \cdot \left( \frac{1}{1-\lambda} \right)\]

Given the above assumptions, the Koyck transformation can be applied to the regression model. The original model is written as: \[y_t = \alpha+\beta_0 \cdot x_t+\beta_0 \cdot \lambda x_{t-1}+\beta_0 \cdot \lambda^2 \cdot x_{t-2}+\dots+\epsilon_t\]

The reformulated equation to be estimated is \[y_t = \alpha \cdot (1-\lambda) + \beta_0 \cdot x_t + \lambda \cdot y_{t-1} + \upsilon_t\] The notation of the error term has changed from $\epsilon_t$ to $\upsilon_t$ in order to highlight that the terms will be different.

koyck = usdata
koyck$year = substr(koyck$date,1,4)
koyck = aggregate(koyck[c("income","consumption")],FUN=sum,by=list(koyck$year))
colnames(koyck) = c("year","income","consumption")
bhat = lm(formula = consumption ~ income + Lag(consumption), data = koyck)
summary(bhat)

## 
## Call:
## lm(formula = consumption ~ income + Lag(consumption), data = koyck)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10648.1   -514.1    260.5    742.7   4689.9 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      -631.29819  574.33825  -1.099    0.275    
## income              0.26835    0.05367   5.000 3.82e-06 ***
## Lag(consumption)    0.71921    0.05906  12.178  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1840 on 73 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.9983, Adjusted R-squared:  0.9982 
## F-statistic: 2.132e+04 on 2 and 73 DF,  p-value: < 2.2e-16