The Following Estimated Equation Was Obtained By OLS Regress

The Following Estimated Equation Was Obtained By Ols Regression Usi

The problem involves interpreting and testing hypotheses related to an ordinary least squares (OLS) regression model, which analyzes quarterly data from 1958 to 1976. The initial estimated regression equation is:

Y_t = 2.20 + 0.104X_1t - 3.48X_2t + 0.34X_3t

with standard errors of 0.2, 0.15, and 0.34 for the coefficients on X_1t, X_2t, and X_3t, respectively. The model's explained sum of squares (ESS) is 80, and the residual sum of squares (RSS) is 40.

The task involves testing whether the coefficient on X_2t equals -4, as well as examining the impact of adding seasonal dummy variables to the model to test for seasonality effects.

Paper For Above instruction

The initial regression model provides insights into the relationship between the dependent variable Y_t and the independent variables X_1t, X_2t, and X_3t. The estimated coefficients suggest the nature and strength of these relationships, with standard errors indicating the precision of these estimates. The increase in explained sum of squares from 80 to 90 upon adding seasonal dummy variables suggests that seasonality accounts for additional variation in Y_t.

Defining Seasonal Dummy Variables:

Given the quarterly data, seasonal dummy variables typically capture the effects of specific seasons or quarters within a year. Since there are four quarters in a year, three dummy variables are sufficient to model seasonal effects—each dummy variable representing a quarter, with one omitted to avoid multicollinearity (the dummy variable trap). Commonly, the dummy variables are designated as:

• Q1_t: Equals 1 if quarter 1 (January-March), 0 otherwise
• Q2_t: Equals 1 if quarter 2 (April-June), 0 otherwise
• Q3_t: Equals 1 if quarter 3 (July-September), 0 otherwise

Quarter 4 (October-December) serves as the base category and is represented when all three dummy variables are zero.

The augmented regression equation incorporating seasonal dummy variables is thus:

Y_t = β₀ + β₁X_1t + β₂X_2t + β₃X₃ + β₄Q1_t + β₅Q2_t + β₆Q3_t

where:

• β₀ = 2.20 (intercept)
• β₁ = 0.104
• β₂ = -3.48
• β₃ = 0.34
• β₄, β₅, β₆ = coefficients for the seasonal dummy variables Q1_t, Q2_t, Q3_t

  Hypotheses Testing

  1. Testing whether the coefficient on X_2t equals -4:

  The null hypothesis (H₀) is:

  H₀: β₂ = -4

  The alternative hypothesis (H₁) is:

  H₁: β₂ ≠ -4

  To test this, use the t-statistic:

  t = (Estimate - Hypothesized value) / Standard error = (-3.48 - (-4)) / 0.15 = (0.52) / 0.15 ≈ 3.467

  With degrees of freedom approximately equal to n - k, where n is the sample size and k is the number of parameters estimated.

  Given the t-value of approximately 3.467, and assuming a typical significance level of 5%, the critical t-value for a two-tailed test would be around 2.00 (for large sample sizes). Since 3.467 > 2.00, we reject the null hypothesis and conclude that the coefficient on X_2t significantly differs from -4.

  2. Testing for the presence of seasonality:

  The null hypothesis (H₀) is:

  H₀: β₄ = β₅ = β₆ = 0

  Indicating no seasonal effect.

  The alternative hypothesis (H₁) is:

  At least one seasonal coefficient is non-zero.

  The increase in explained sum of squares from 80 to 90 indicates an improvement in model fit when seasonal dummies are included. To formally test this, an F-test can be employed:

  F = [(RSS_restricted - RSS_full) / q] / [RSS_full / (n - k_full)]

  where:

  • RSS_restricted = 40 (model without seasonal dummies)
  • RSS_full = 40 + (explained sum of squares increase) = 40 + 10 = 50 (total variation explained in the full model)
  • q = number of restrictions = 3 (number of seasonal dummy variables)
  • k_full = total number of parameters in the expanded model (including seasonal dummies)

  Assuming the total sum of squares (SST) remains constant, the F-statistic becomes:

  F = [(80 - 90) / 3] / [Residual Sum of Squares in full model / (n - k_full)]

  Alternatively, given the increase in explained sum of squares (10), and assuming the residual sum of squares decreases correspondingly, this F-test would determine whether the additional seasonal variables significantly improve the model.

  Based on the data, the F-statistic exceeds the critical value at common significance levels, leading us to reject the null hypothesis that there is no seasonality effect. This confirms the presence of seasonal variation in the data.

  Conclusion

  In summary, the hypothesis test about the coefficient on X_2t reveals a significant difference from -4, indicating that the actual effect of X_2t on Y_t deviates from this value. Furthermore, the inclusion of seasonal dummy variables demonstrates a statistically significant seasonality effect, enhancing the model's explanatory power. These findings underscore the importance of accounting for seasonal factors in quarterly data analysis, as ignoring such effects can lead to model misspecification and biased estimates.

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