Estimate The Following Equation By OLS Using Ordinary Least

Estimate The Following Equation By Ols Using Ordinary Least Squares

Estimate the following equation by OLS using ordinary least squares: 01 tttpq =++ββε where and . ln() ttpP = ln(Q) ttq = 2. What is your estimate of the elasticity of demand? Your estimate is positive, so it looks like something is wrong. In the following questions, we will try to figure out what is wrong.

First, the value of a dollar changed a lot from 1926 to 2014. We should really use real prices rather than nominal prices. Run the regression: , 01 tttrq =++ββε where is the real price. ln()ln(CPI) tttrP =−

Compute a 95% confidence interval for β1.

Test for autocorrelation in the errors of your regression in (3). What are the implications of your test result for interpreting your results in (3) and (4)?

Use the Newey-West correction to fix the regression in (3). Try using up to 12 lags in the correction.

Plot the log real price (rt) over time. What is the long-run trend in prices?

It is possible that prices are being driven by some trends unrelated to quantity. Re-estimate your regression model in (3) with the year as an additional right-hand-side variable.

Now, let’s consider changing the model by adding the lags of price and quantity. 012131 tttttrqrq −−=++++ββββε Test for autocorrelation.

Using the discussion on slides 21 and 23 from Ch 9, interpret the results from your regression in (9). What is the long-run elasticity of demand? Interpret the error correction model.

We are interpreting our regression parameters as inverse elasticities of demand. What precisely are we assuming about how corn production is determined?

Paper For Above instruction

Introduction

The estimation of demand elasticity forms a crucial part of understanding market dynamics, policy impacts, and economic forecasting. The specific case of demand estimation using ordinary least squares (OLS), and the subsequent refinement of the model by accounting for price definitions, autocorrelation, trend effects, and lag structures, provides a comprehensive methodology to uncover true elasticities and causal relationships. This paper discusses the steps, corrections, and interpretations involved in estimating demand elasticity, using real prices, autocorrelation tests, confidence intervals, and integrated models, with a particular focus on the demand for corn as a case study.

Estimating Demand: Basic OLS Model

The initial step involves estimating the demand equation: \(q_t = \beta_0 + \beta_1 p_t + \varepsilon_t\), where \(q_t\) is the quantity demanded and \(p_t\) is the price at time \(t\). By taking logs, the model becomes \(\ln(q_t) = \beta_0 + \beta_1 \ln(P_t) + \varepsilon_t\). The elasticity of demand is estimated by \(\beta_1\). However, initial estimates often yield positive \(\beta_1\), contrary to economic theory, which suggests demand should respond negatively to price increases. This anomaly indicates potential issues such as non-stationarity, measurement inaccuracies, or omitted variable bias.

Adjusting for Price Changes over Time

Recognizing that nominal prices change due to inflation, it's essential to convert nominal prices into real prices. Using the Consumer Price Index (CPI), the real price is calculated as \(\ln(P_{real,t}) = \ln(P_{nominal,t}) - \ln(CPI_t)\). Re-estimating the demand function with real prices typically yields more theoretically consistent estimates. Computing a 95% confidence interval for \(\beta_1\) involves using the standard error of the estimate and the t-distribution, providing bounds within which the true elasticity likely resides.

Testing for Autocorrelation and Applying Corrections

Autocorrelation in the residuals can invalidate standard errors and confidence intervals, misleading inference. The Durbin-Watson test often detects autocorrelation; if present, applying the Newey-West correction adjusts standard errors to account for serial correlation. Using up to 12 lags captures autocorrelation patterns over multiple periods, improving the robustness of inference about the elasticity.

Analyzing Price Trends Through Visualization

Plotting the log real prices over time reveals long-term trends, which may be non-stationary, indicating the presence of persistent upward or downward movements driven by technological changes, policy shifts, or macroeconomic factors. Recognizing these trends suggests the need to control for them in regressions to avoid spurious results. Including a time trend or the year variable as an additional regressors helps isolate the true demand response from secular price movements.

Incorporating Lagged Variables and Error Correction

Extending the model to include lagged prices and quantities, such as in the error correction model (ECM): \(q_t = \beta_0 + \beta_1 p_t + \beta_2 q_{t-1} + \beta_3 p_{t-1} + \varepsilon_t\), captures dynamic adjustments and potential long-run relationships. Testing for autocorrelation in this extended model informs about the stability of the demand system. The interpretation of the estimated elasticities in this context involves understanding the long-run responsiveness of demand, considering both immediate and lagged effects.

Interpretation and Assumptions

The estimated elasticities are treated as inverse relationships under the assumption that demand is primarily driven by price and quantity changes without significant external shocks. The model assumes that corn production is driven by factors captured within the demand and supply framework, with prices adjusting to meet equilibrium. Critical assumptions also include rational expectations and market clearing conditions, ensuring that the estimated parameters reflect true elasticities rather than artifacts of omitted variables or measurement errors.

Conclusion

Estimating demand elasticity through OLS requires careful attention to data issues, model specification, and statistical corrections. Adjusting for inflation, testing and correcting autocorrelation, and including trends and lag structures substantially improve the reliability of estimates. The long-run elasticity derived from the extended models provides valuable insights into market responsiveness, guiding policymakers and economists in understanding the dynamics of commodity demand, particularly in agricultural markets like corn.

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