Refer To Example 56 In The Chapter It Was Shown That The Per

529refer To Example 56 In The Chapter It Was Shown That The Percen

Refer to Example 5.6 in the chapter. It was shown that the percentage change in the index of hourly earnings and the unemployment rate from the textbook followed the traditional Phillips curve model. An updated version of the data, from the textbook, can be found here as Table 5-19, and on the textbook’s website. Create a scattergram using the percentage change in hourly earnings as the Y variable and the unemployment rate as the X variable.

Does the graph appear linear? Now create a scattergram as above, but use 1/X as the independent variable. Does this seem better than the graph in part (a)? Fit Equation (5.29) to the new data. Does this model seem to fit well?

Also create a regular linear (LIV) model as in Equation (5.30). Which model is better? Why?

Paper For Above instruction

The Phillips curve is a fundamental concept in macroeconomics, illustrating the inverse relationship between unemployment rates and inflation or wage changes. The traditional Phillips curve suggests that lower unemployment rates are associated with higher inflationary pressures, and vice versa. In examining the relationship between the percentage change in hourly earnings and unemployment rates, this analysis seeks to assess whether this classical inverse correlation persists or if alternative functional forms better describe the data.

Using the updated data from Table 5-19, which aligns with the example in the chapter, the initial step involves constructing a scattergram plotting the percentage change in hourly earnings (Y variable) against the unemployment rate (X variable). Visually inspecting this scatterplot allows us to evaluate whether a linear relationship exists between these two variables. If the points seem to follow a trend that approximates a straight line, a linear model may be appropriate. Typically, in the context of the Phillips curve, a negative linear relationship is hypothesized, reflecting the trade-off between unemployment and wage inflation.

Upon plotting the scattergram, if the data points do not align closely along a straight line, the relationship may be nonlinear or influenced by other factors. To test for this, a second scattergram is created by transforming the independent variable to 1/X (the reciprocal of the unemployment rate). This transformation is grounded in the hypothesis that the relationship between wage change and unemployment rate might be better approximated by a hyperbolic or inverse function rather than a linear one. The scatterplot of percentage change in hourly earnings against 1/X can visually reveal whether this transformation improves the linearity of the relationship.

If the scattergram with 1/X as the independent variable appears more linear, the next step involves fitting Equation (5.29), which models the relationship as:

Y = β₀ + β₁ * (1/X) + ε

This model assumes a linear relationship between the wage change and the reciprocal of the unemployment rate. The statistical fit of this model can be evaluated through R-squared values, residual analysis, and significance tests of the coefficients. A high R-squared and statistically significant coefficients suggest that the model captures the relationship effectively.

In addition to this nonlinear model, a traditional linear (LIV) model as in Equation (5.30) can be estimated, which regresses the percentage change in hourly earnings directly on the unemployment rate:

Y = α + γX + ε

This linear model provides a baseline against which to compare the transformed model's fit. The comparison involves examining statistical measures such as the adjusted R-squared, visual residual plots, and the significance levels of the regression coefficients.

The model that better captures the relationship between wage change and unemployment rate—evidenced by higher explanatory power, better residual diagnostics, and theoretical consistency—can be considered superior. Given the historical context of the Phillips curve, the inverse (1/X) version often aligns better with empirical data, especially if the scattergram indicates non-linearity in the original formulation. However, the linear model's simplicity and ease of interpretation often make it a default choice unless the transformation yields significant improvements.

In conclusion, assessing the fit of these models through graphical and statistical methods reveals whether the classical inverse relationship holds or whether adjustments, such as the reciprocal transformation, offer a more accurate depiction. These insights are vital in macroeconomic policy discussions, particularly concerning wage-setting, inflation expectations, and unemployment dynamics.

References

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