Time Series Regression Analysis

Dynamic Effects and Distributed Lag Models

Time series regression is a statistical method used to analyze data points collected or recorded at specific time intervals. In this context, the focus is on understanding how cold weather events (measured as Freezing Degree Days, FDD) affect orange juice prices over time using distributed lag models.

• Distributed Lag Regression: A model that relates the current value of a dependent variable to current and past values of an independent variable.

• Dynamic Causal Effects: The impact of an event (e.g., a freeze) on a variable (e.g., price) over several subsequent periods.

• Exogeneity: FDD is considered exogenous because weather cannot be influenced by human actions, making it suitable for causal inference.

Equation (Distributed Lag Model):

Where:

• \%ChgPt: Percentage change in orange juice price in month t

• FDDt: Number of freezing degree days in month t

• ut: Error term

Estimation and Standard Errors

Ordinary Least Squares (OLS) is used to estimate the coefficients. Because time series data often exhibit serial correlation in errors, Heteroskedasticity and Autocorrelation Consistent (HAC) standard errors (Newey–West) are applied. The truncation parameter m is chosen based on sample size:

For 612 monthly observations, .

Dynamic Multipliers and Cumulative Effects

Interpretation of Dynamic Multipliers

Dynamic multipliers measure the effect of a unit increase in FDD on orange juice prices in subsequent months. Cumulative multipliers sum these effects over time.

• Immediate Effect: A single freezing degree day increases prices by 0.50% in the same month.

• Lagged Effects: The effect persists but diminishes over subsequent months (e.g., 0.17% after one month, 0.07% after two months).

• Cumulative Effect: The total effect accumulates, peaking around the seventh month.

Table: Dynamic and Cumulative Multipliers (Selected Lags)

Additional info: Standard errors are in parentheses. The cumulative multiplier after 18 months is not statistically significant at the 10% level.

Statistical Significance and Confidence Intervals

• Dynamic multipliers are plotted with 95% confidence intervals ( HAC standard errors).

• Only the initial effect is statistically significant at the 5% level; subsequent effects are not.

Sensitivity Analysis

Robustness to HAC Truncation Parameter

Changing the HAC truncation parameter (e.g., from to ) does not materially affect the results, indicating robustness.

Omitted Variable Bias and Seasonality

• Seasonal demand for orange juice could correlate with FDD, potentially causing omitted variable bias.

• Including monthly indicator variables (seasonal dummies) in the regression does not significantly change the results.

• These indicators are not jointly significant at the 10% level (), suggesting seasonality is not a major source of bias.

Stability of Dynamic Multipliers Over Time

Testing for Stability

The stability of regression coefficients is tested using the Quandt Likelihood Ratio (QLR) statistic:

• QLR statistic for the full model: 21.19 (with 20 degrees of freedom).

• 1% critical value: 2.43. The hypothesis of stability is rejected at the 1% significance level.

• Subsample analysis shows the effect of freezes on prices was larger and more persistent in the 1950s–1960s, diminished in the 1970s, and became less persistent in the 1980s–1990s.

Example: Subsample Cumulative Multipliers

• 1950–1966: Large, persistent effect.

• 1967–1983: Diminished but persistent effect.

• 1984–2000: Short-run effect similar to previous period, but persistence is much reduced.

Advanced Estimation Methods: ADL and GLS

Strict Exogeneity and Consistency

• GLS and ADL models can be more efficient if the error term is serially correlated and FDD is strictly exogenous.

• Strict exogeneity requires the error term to have mean zero given past, present, and future values of FDD.

• Because traders use weather forecasts, the error term may be correlated with future FDD, violating strict exogeneity.

• Therefore, GLS and ADL estimators are not used in this application.

Applications and Insights

Market Adaptation and Nonstationarity

• Changes in the orange juice market, such as the migration of orange groves to southern Florida, have reduced the impact of northern freezes on prices.

• Nonstationarity refers to changes in the underlying data-generating process over time.

Forecasting vs. Causal Effects

• Orange juice futures prices are effective predictors of cold weather but do not cause temperature changes.

• Excess volatility in futures prices may exist, meaning price movements are not fully explained by fundamentals.

Summary Table: Key Concepts and Terms

Additional info: The notes expand on distributed lag models, dynamic multipliers, and time series regression, providing context for business-statistics students.