Question

The file FISH contains 97 daily price and quantity observations on fish prices at the Fulton Fish Marke in New York City. Use the variable log(avgprc) as the dependent variable. $\begin{array}{l}\text{ (i) Regress log(avgprc) on four daily dummyvariables, with Friday as the base. Include a linear }\\ \text{ timetrend. Is there evidence that price varies systematically within a week? }\end{array}$ $\begin{array}{l}\text{ (ii) }\text{ Now, add the variables wave }2\text{ and wave, which are measures of wave heights over the past }\\ \text{ several days. Are these variables individually significant? Describe amechanism by which }\\ \text{ stormy seas would increase the price offish. }\end{array}$ $\begin{array}{l}\text{ (iii) What happened to the time trend when wave }2\text{ and wave }3\text{ were added to the regression? What }\\ \text{ must be going on? }\end{array}$ $\begin{array}{l}\text{ (iv) Explain why all explanatory variables in theregression are safely assumed to be strictly }\\ \text{ exogenous. }\\ \text{ (v) Test the errors for AR (1) serial correlation. }\end{array}$ $\begin{array}{l}\text{ (vi) Obtain the Newey-West standard errors usingfour lags. What happens to the }t\text{ statistics on }\\ \text{ wave }2\text{ and waves? Did you expect a bigger or smaller change compared withthe usual OLS }\\ t\text{ statistics? }\\ \text{ (vii) Now, obtainthe Prais-Winsten estimates for the model estimated in part (ii). Are wave?and }\\ \text{ wave }3\text{ jointly statistically significant? }\end{array}$

Short answer

Prices appear to vary systemically weekly; high waves can raise prices by limiting supply; wave effects reduce with Newey-West errors and are significant jointly in Prais-Winsten.

Step-by-step solution

Log Transformation and Regression with Dummies

First, we take the natural logarithm of the average price `avgprc` to use it as the dependent variable in the regression. We then regress `log(avgprc)` on four daily dummy variables representing Monday to Thursday, with Friday as the base category. We also include a linear time trend to account for underlying temporal changes in price. Perform a hypothesis test on the coefficients of the dummies to check if there's systematic variation in prices across different days of the week.

Add Wave Variables

In this step, include the variables `wave2` and `wave3`—which represent wave heights of past days—into the regression from Step 1. Test each wave variable for significance using a t-test on their coefficients. A rationale for wave variables impacting price is attributed to supply chain disruption; larger waves would hinder fishing activities, thus reducing fish supply and potentially increasing prices.

Analyze Changes in Time Trend

Compare the coefficient and significance of the linear time trend before and after adding `wave2` and `wave3` to the model. If the significance or coefficient of the time trend changes, it might indicate that wave heights are correlated with an underlying pattern or omitted variable previously captured by the time trend.

Exogeneity of Explanatory Variables

Discuss why the explanatory variables are assumed to be strictly exogenous. Given that these variables include time-invariant characteristics (day of the week dummies) and natural phenomena (wave heights), they are unlikely influenced by past prices, meaning they meet the condition for strict exogeneity.

Test for Serial Correlation

Use the Durbin-Watson test or Breusch-Godfrey test to examine the residuals from the regression for AR(1) serial correlation. This involves testing whether past error terms are correlated with current error terms, which would violate Ordinary Least Squares (OLS) assumptions.

Compute Newey-West Standard Errors

Adjust the standard errors of the regression using Newey-West to account for heteroscedasticity and serial correlation in the residuals, applying four lags. Determine how the adjusted t-statistics for `wave2` and `wave3` compare to those from the original OLS regression, noting that they usually become smaller with this correction.

Prais-Winsten Estimation

Re-estimate the model from Step 2 using Prais-Winsten estimation, which corrects for serial correlation within the data. Evaluate whether the wave variables are jointly significant by conducting an F-test. Compare these results to your earlier findings to check for consistency in coefficient significance and model robustness.

Key concepts

Dummy Variables

Dummy variables are artificial variables created to represent an attribute with two or more distinct categories/values. These variables take on the value of zero or one. In the context of the econometrics exercise with fish prices, each day of the week (Monday to Thursday) is transformed into a dummy variable. This is done by assigning a "1" to represent its presence and a "0" for its absence in a particular observation.

The purpose of using dummy variables in this exercise is to ascertain if there are systematic variations in fish prices across different weekdays. In this case, Friday is chosen as the base category. That means all differences in price are measured relative to Friday.

• Dummy variables help identify the impact of categorical attributes.
• They are essential for using qualitative data in regression models.
• By interpreting the coefficients of these dummy variables, you can understand the effect of different weekdays on price fluctuations.

Time Series Analysis

Time series analysis is a statistical technique that deals with data points collected or recorded at specific time intervals. It is used to identify trends, cycles, and seasonal variations to make predictions or understand past behavior. In the given econometric exercise, a linear time trend is incorporated to detect temporal changes in fish prices over the observed period.

By including a time trend, economists can separate time-dependent patterns from other explanatory factors. This distinction is crucial when determining whether changes are due to inherent time dynamics or other variables, like significant wave heights.

• Time series models allow examination of daily, monthly, or yearly data for patterns or changes.
• These models are essential for forecasting and policy formulation based on historical data trends.
• Detecting trend changes helps differentiate between temporary disruptions and long-term shifts.

Serial Correlation

Serial correlation, or autocorrelation, occurs when residual errors in a regression model are correlated with one another. This violates one of the key assumptions of ordinary least squares (OLS) regression, where error terms should be independent. In this exercise, testing for AR(1) serial correlation is important to ensure the reliability of the regression results. Methods like the Durbin-Watson test can detect whether there is correlation among residuals from one day to the next in the fish price model.

• Serial correlation can lead to inefficient estimates and underestimated standard errors.
• Autocorrelation suggests that a past error impacts the present error, affecting the model’s predictive power.
• Correcting for serial correlation improves the model's accuracy and the validity of inferences.

Exogenous Variables

Exogenous variables are those that are determined outside of the model and are not influenced by the dependent variable. In this econometrics exercise, explanatory variables like the day of the week and wave heights are considered exogenous. They are assumed to affect fish prices without being influenced by past or future prices.

The rationale behind the assumption of exogeneity is grounded in the nature of these variables:

• Day of the week is time-invariant and unlikely affected by fish prices.
• Wave heights are natural phenomena, independent of market conditions.
• Exogeneity is crucial for unbiased and consistent estimation of the impacts of independent variables on the dependent variable.

Standard Errors

Standard errors measure the extent to which the sample estimate of a coefficient is expected to vary from the actual population parameter. They are crucial for reliable hypothesis testing in econometrics.
In this exercise's context, the Newey-West standard errors are used to correct for potential issues like heteroskedasticity and autocorrelation in the residuals. By applying this correction, we get more robust t-statistics for understanding the significance of explanatory variables in the regression model.

• Corrected standard errors provide more accurate and realistic confidence intervals for estimates.
• Newey-West is a method used specifically when residuals exhibit time-related dependence.
• Adjusted standard errors ensure that hypothesis tests do not produce misleading conclusions due to violation of OLS assumptions.