Estimation of Dynamic Causal Effects in Time Series Regression

Estimation of Dynamic Causal Effects

Introduction to Dynamic Causal Effects

Dynamic causal effects describe how a change in one variable (X) affects another variable (Y) over time. In business and economics, understanding these effects is crucial for forecasting and policy analysis. The distributed lag regression model is a primary tool for estimating such effects, allowing Y to depend on current and past values of X.

• Dynamic causal effect: The sequence of impacts on Y following a change in X, measured over multiple time periods.

• Distributed lag model: A regression model where Y is a function of X and its lagged values.

• Example: The effect of freezing weather in Florida on orange juice prices, where both immediate and delayed impacts are considered.

Distributed Lag Regression Model

Model Specification

The distributed lag model expresses the outcome variable as a function of current and past values of the explanatory variable:

• General form:

$Y_t = \beta_0 + \beta_1 X_t + \beta_2 X_{t-1} + \beta_3 X_{t-2} + \cdots + \beta_{r+1} X_{t-r} + u_t$

• Dynamic multipliers: The coefficients $\beta_1, \beta_2, \ldots, \beta_{r+1}$ represent the effect of a unit change in X at different lags.

• Cumulative dynamic multipliers: The sum of dynamic multipliers up to a given lag, representing the total effect over time.

• Example: In the orange juice case, a freeze today increases prices immediately and in subsequent months.

Exogeneity and Strict Exogeneity

For valid estimation of dynamic causal effects, the regressors must be exogenous:

• Exogeneity: The error term has a conditional mean of zero given current and past values of X.

• Strict exogeneity: The error term has a conditional mean of zero given past, present, and future values of X.

Key Concept Table:

Type	Definition
Exogeneity	$E(u_t | X_t, X_{t-1}, X_{t-2}, \ldots) = 0$
Strict Exogeneity	$E(u_t | \ldots, X_{t+2}, X_{t+1}, X_t, X_{t-1}, \ldots) = 0$

Additional info: Strict exogeneity is required for more efficient estimation methods like GLS and ADL models.

Estimation Methods

Ordinary Least Squares (OLS) Estimation

When X is exogenous, OLS can be used to estimate the distributed lag model. The assumptions for valid OLS estimation are:

• X is exogenous.

• Variables are stationary and become independent as time separation increases.

• Large outliers are unlikely (more than eight finite moments).

• No perfect multicollinearity among regressors.

Autocorrelation and HAC Standard Errors

Time series errors are often autocorrelated, making standard OLS errors unreliable. HAC (heteroskedasticity- and autocorrelation-consistent) standard errors adjust for this:

• Newey–West estimator: A widely used HAC estimator, with truncation parameter $m$ chosen as $m = 0.75 T^{1/3}$.

• Formula for variance adjustment:

$\tilde{\sigma}_{\hat{\beta}_1}^2 = \hat{\sigma}_{\hat{\beta}_1}^2 \hat{f}_T$

• Truncation parameter: Determines how many autocorrelations are included in the variance estimate.

Dynamic Multipliers and Cumulative Multipliers

Dynamic multipliers measure the effect of a unit change in X at each lag. Cumulative multipliers sum these effects over time:

• Dynamic multiplier (h-period): $\beta_{h+1}$

• Cumulative multiplier (h-period): $\beta_1 + \beta_2 + \cdots + \beta_{h+1}$

• Long-run cumulative multiplier: $\beta_1 + \beta_2 + \cdots + \beta_{r+1}$

Alternative Estimation: ADL and GLS

If X is strictly exogenous, more efficient estimators are available:

• ADL (Autoregressive Distributed Lag) Model: Incorporates lagged values of Y and X, making errors serially uncorrelated.

• GLS (Generalized Least Squares): Transforms the model to remove autocorrelation, then estimates by OLS.

• Feasible GLS: Uses estimated autocorrelation parameters (e.g., Cochrane–Orcutt method).

Additional info: These methods require strict exogeneity; otherwise, estimates may be inconsistent.

Application: Orange Juice Prices and Cold Weather

Empirical Analysis

Monthly data from 1950–2000 on orange juice prices and freezing degree days in Florida are used to estimate dynamic causal effects. The distributed lag regression model is applied:

• Regression equation:

$\hat{\%ChgP}_t = -0.65 + 0.47 FDD_t + 0.14 FDD_{t-1} + 0.06 FDD_{t-2} + \cdots$

• Interpretation: Each freezing degree day increases prices by 0.47% in the current month, 0.14% in the next month, etc.

• HAC standard errors: Used to account for autocorrelation in errors.

Table: Dynamic and Cumulative Multipliers

Main purpose: To show the effect of a freezing degree day on orange juice prices over time.

Lag Number	Dynamic Multipliers	Cumulative Multipliers
0	0.50 (0.14)	0.50 (0.14)
1	0.17 (0.09)	0.67 (0.14)
2	0.07 (0.06)	0.74 (0.17)
3	0.07 (0.04)	0.81 (0.18)
4	0.02 (0.03)	0.84 (0.19)
5	0.03 (0.03)	0.87 (0.19)
6	0.03 (0.05)	0.90 (0.20)
12	-0.14 (0.08)	0.54 (0.27)
18	0.00 (0.02)	0.37 (0.30)

Additional info: Standard errors are in parentheses. The cumulative effect peaks at 7 months, then declines.

Sensitivity and Stability

• Results are robust to changes in HAC truncation parameter.

• Including monthly indicators for seasonality does not affect results.

• Stability of dynamic multipliers over time is tested using the QLR statistic; results show significant changes over decades.

Exogeneity in Practice: Examples

When Is Exogeneity Plausible?

Exogeneity depends on context and economic theory. Examples:

• U.S. income and Australian exports: U.S. income is plausibly exogenous.

• Oil prices and inflation: Oil prices are endogenous due to strategic decisions by producers.

• Monetary policy and inflation: Interest rates are endogenous, set in response to economic conditions.

• GDP growth and term spread: Term spread is not strictly exogenous due to simultaneous determination with GDP.

Conclusion

Estimating dynamic causal effects in time series requires careful consideration of exogeneity. OLS estimation of distributed lag models is valid when regressors are exogenous. For strictly exogenous regressors, more efficient methods like ADL and GLS are available. In practice, economic theory and institutional knowledge are essential for assessing exogeneity and choosing appropriate estimation methods.