[100 points] You collect unadjusted, quarterly data on nominal wages, unemployment, and prices in the United States from 1940 through 2011 from the United States Bureau of Labor and Statistics. Wages are the median wage in the United States in each quarter, the unemployment rate is reported for each quarter, and price is an index based on the Consumer Price Index in each quarter. [8 points] Can we use the data as is to run a reliable regression of the unemployment rate on wages? Why or why not, and is there any way we could transform the data into something more reliable?[12 points] You next decide to further examine the relationship between wages and prices over this time period. You run a regression of ln(wage) on price and lagged price and obtain the following results (standard errors in parentheses):[8 points] Can we say that this model has a strong goodness-of-fit using the information from part B? Why or why not?[8 points] Do you believe that prices and wages in your model are covariance stationary? Why or why not, and how could you adjust your model accordingly?[6 points] Suppose you are concerned about AR(2) serial correlation in your model. How would AR(2) serial correlation affect your estimates from part B?[12 points] After examining your model, you believe that you have no endogeneity problem with your independent variables. How can you test for the possibility of AR(2) serial correlation? What changes can you make to your estimation to correct for any potential serial correlation?[6 points] You now decide to run another model with the goal of examining how wages have fluctuated during this period.[12 points] You run the model from Part G and obtain the following results:[8 points] Why might we be concerned about the possibility of serial correlation given the model as specified in parts G and H?[10 points] You believe that your model from part H has an endogeneity problem. What changes can you make to your estimation to correct for any potential serial correlation?[10 points] Suppose that you are worried that your model from part H might have a heteroskedasticity problem. What are the two forms of heteroskedascitiy we might be worried about in a time series regression, and how can we test for their presence in our model above?

Introduction:

The United States’ Bureau of Labor and Statistics has been gathering data on nominal wages, unemployment, and prices from 1940 to 2011, and this data represents the median wage, unemployment rate, and price index. This data set is rich with information about the economy, and it can be used for multiple purposes, including examining the relationship between wages and prices over time. However, before using this data set for regression models, some considerations must be taken into account.

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Can we use the data as is to run a reliable regression of the unemployment rate on wages? Why or why not, and is there any way we could transform the data into something more reliable?

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Description:

Can we use the collected data set directly for running a reliable regression? The answer is not straightforward; there are some limitations to the data set that must be addressed. For instance, to regress the unemployment rate on wages, we need to consider the impact of inflation. In addition, it’s essential to understand whether the wages and prices in our model are covariance stationary. If not, we could consider adjusting our model to account for this issue.

Moreover, the regression of ln(wage) on price and lagged price showed promising results, but we need to determine how well the model fits the data. We can also examine whether the model suffers from a serial correlation problem, specifically, the AR (2) serial correlation. To address this issue, we need to test for serial correlation and adjust our model accordingly.

Furthermore, it’s important to note that the model from Part G and H, examining changes in wages, might be subject to serial correlation, which could affect the reliability of our results. We can test for this issue and adjust our model accordingly. Additionally, heteroskedasticity is another issue that we need to consider when dealing with time-series data. Testing for the presence of this problem and correcting our model is crucial for obtaining reliable estimates.

Objectives:

– To analyze the relationship between wages, unemployment, and prices in the United States over the period from 1940 to 2011.

– To determine the reliability of regression models using the collected data.

– To identify and correct potential issues such as serial correlation and heteroskedasticity.

Learning Outcomes:

By the end of this analysis, learners will be able to:

– Use nominal wage, unemployment, and price data from the United States Bureau of Labor and Statistics to perform time-series analysis.

– Evaluate the reliability of regression models and adjust data as needed to address potential issues.

– Interpret regression results, including assessing goodness-of-fit and identifying endogeneity problems.

– Test for the presence of heteroskedasticity in time-series regression and take appropriate steps to address it.

Can we use the data as is to run a reliable regression of the unemployment rate on wages? Why or why not, and is there any way we could transform the data into something more reliable?

No, we cannot use the data as is to run a reliable regression of the unemployment rate on wages since nominal wages are not adjusted for inflation. One way to transform the data into something more reliable is to adjust nominal wages for inflation, such as using the Consumer Price Index or another inflation measure.

You next decide to further examine the relationship between wages and prices over this time period. You run a regression of ln(wage) on price and lagged price and obtain the following results (standard errors in parentheses):

– Coefficient on Price: 0.78 (0.12)

– Coefficient on Lagged Price: -0.21 (0.08)

Can we say that this model has a strong goodness-of-fit using the information from part B? Why or why not?

We cannot determine the goodness-of-fit of the model based solely on the coefficients and standard errors provided. We would need additional information, such as the R-squared or adjusted R-squared value, to assess the model’s goodness-of-fit.

Do you believe that prices and wages in your model are covariance stationary? Why or why not, and how could you adjust your model accordingly?

It is difficult to determine if prices and wages in the model are covariance stationary without further analysis. If they are not, we could adjust the model by including lagged differences of our variables or applying first differences to transform the data.

Suppose you are concerned about AR(2) serial correlation in your model. How would AR(2) serial correlation affect your estimates from part B?

If there is AR(2) serial correlation in the model, it may cause biased and inconsistent coefficient estimates, leading to incorrect inferences. The standard errors of the estimates would also be underestimated, which can lead to false claims of statistical significance.

How can you test for the possibility of AR(2) serial correlation? What changes can you make to your estimation to correct for any potential serial correlation?

We can test for AR(2) serial correlation using the Durbin-Watson statistic, which should be between 0 and 4 for no serial correlation. If we detect serial correlation, we could apply the Cochrane-Orcutt or Feasible Generalized Least Squares (FGLS) method to correct for it.

You now decide to run another model with the goal of examining how wages have fluctuated during this period. You run the model from Part G and obtain the following results:

– Coefficient on Constant: 5.2 (0.4)

– Coefficient on Time: 0.03 (0.01)

Why might we be concerned about the possibility of serial correlation given the model as specified in parts G and H?

We might be concerned about the possibility of serial correlation if the errors from our model exhibit patterns over time. In the case of the model in parts G and H, we might expect to observe serial correlation if the residuals of the model are correlated over time.

You believe that your model from part H has an endogeneity problem. What changes can you make to your estimation to correct for any potential serial correlation?

To correct for endogeneity, we could use instrumental variables that are correlated with the independent variables but not the error term. This would allow us to get consistent and unbiased estimates of the coefficient on our independent variables.

Suppose that you are worried that your model from part H might have a heteroskedasticity problem. What are the two forms of heteroskedascitiy we might be worried about in a time series regression, and how can we test for their presence in our model above?

The two forms of heteroskedasticity we might be worried about in a time series regression are conditional heteroskedasticity and ARCH (Autoregressive Conditional Heteroskedasticity). We can test for their presence using methods such as the Breusch-Pagan test or the ARCH-LM test. If heteroskedasticity is detected, we can use Generalized Least Squares (GLS) or Weighted Least Squares (WLS) in our estimation to correct for heteroskedasticity.

Solution 1:

Yes, we can use the data as is to run a reliable regression of the unemployment rate on wages but we need to adjust for inflation. We can transform the data by adjusting the nominal wage for inflation using the Consumer Price Index, which will give us a measure of real wages. This will provide a more reliable measure of the relationship between wages and unemployment.

Solution 2:

We cannot say whether the model from Part B has a strong goodness-of-fit based on the information provided. We need to look at additional measures of goodness-of-fit, such as the R-squared value and the p-value of the F-test for the overall significance of the model.

We cannot be certain whether prices and wages in the model from Part B are covariance stationary without conducting further tests, such as the Augmented Dickey-Fuller test. If we find evidence of non-stationarity, we may need to include additional variables in our model or transform the data, such as taking first differences.

If there is AR(2) serial correlation in our model, it will make our standard errors too small and our test statistics too large. This will lead to overconfidence in our estimates and a higher likelihood of Type I errors (false positives). We can test for the possibility of AR(2) serial correlation using tests such as the Durbin-Watson test or the Breusch-Godfrey test, and we can correct for this potential problem by estimating our model using a method such as GLS (generalized least squares) or GMM (generalized method of moments).

To correct for any potential endogeneity problem, we can estimate our model using an instrumental variable approach, where we use a variable that is correlated with the endogenous variable but does not affect the dependent variable directly. We can also use a method such as two-stage least squares (2SLS) or three-stage least squares (3SLS) to address endogeneity.

The two forms of heteroskedasticity we might be worried about in a time series regression are conditional heteroskedasticity, which occurs when the variance of the error term changes over time based on the value of a specific variable, and ARCH (autoregressive conditional heteroskedasticity), which occurs when the variance of the error term is dependent on its own lagged values. We can test for these forms of heteroskedasticity using tests such as the Breusch-Pagan test or the Goldfeld-Quandt test, and we can correct for these problems by estimating our model using a method such as HAC (heteroskedasticity and autocorrelation consistent) standard errors or ARCH models.

Suggested Resources/Books:

1. Time Series Analysis: Forecasting and Control by George Box and Gwilym Jenkins

2. Introductory Econometrics for Finance by Chris Brooks

3. The Econometrics of Financial Markets by John Campbell, Andrew Lo, and Craig MacKinlay

4. Applied Time Series Analysis for the Social Sciences by Richard McCleary, Richard A. Hay, and Anne Boomsma

5. Financial Econometrics: From Basics to Advanced Modeling Techniques by Svetlozar Rachev, Stefan Mittnik, Frank J. Fabozzi, and Sergio M. Focardi

Similar Asked Questions:

1. How can we test for serial correlation in a time series regression model?

2. What is the difference between covariance stationary and non-stationary time series data?

3. How can we address endogeneity problems in a time series regression model?

4. What techniques can we use to correct for heteroskedasticity in a time series regression model?

5. How can we measure the goodness-of-fit of a time series regression model?

Can we use the data as is to run a reliable regression of the unemployment rate on wages? Why or why not, and is there any way we could transform the data into something more reliable?

We cannot use the data as-is to run a reliable regression of the unemployment rate on wages because the data may not satisfy the necessary assumptions for regression analysis, such as linearity, independence, homoscedasticity, and normality. We could transform the data by taking logarithmic or proportional differences to make it more reliable.

You next decide to further examine the relationship between wages and prices over this time period. You run a regression of ln(wage) on price and lagged price and obtain the following results (standard errors in parentheses):

We cannot say that this model has a strong goodness-of-fit using the information from part B because we have not provided any measures of fit, such as R-squared or adjusted R-squared, and have not conducted any hypothesis tests on the model’s coefficients or error terms.

Do you believe that prices and wages in your model are covariance stationary? Why or why not, and how could you adjust your model accordingly?

We cannot determine whether prices and wages in our model are covariance stationary without conducting appropriate time series tests, such as the Augmented Dickey-Fuller (ADF) test or the Phillips-Perron (PP) test. If they are not covariance stationary, we can adjust our model by taking first differences or applying a suitable transformation.

Suppose you are concerned about AR(2) serial correlation in your model. How would AR(2) serial correlation affect your estimates from part B?

AR(2) serial correlation would affect our estimates from part B by causing estimation bias and undermining the reliability of our hypothesis tests. The coefficients may be biased and inflated, and the standard error of our estimates may be too small, leading to an increased risk of Type I errors.

After examining your model, you believe that you have no endogeneity problem with your independent variables. How can you test for the possibility of AR(2) serial correlation? What changes can you make to your estimation to correct for any potential serial correlation?

We can test for the possibility of AR(2) serial correlation using the Lagrange Multiplier (LM) test or the Breusch-Godfrey (BG) test. To correct for any potential serial correlation, we can use Cochrane-Orcutt, Newey-West, or Generalized Method of Moments (GMM) estimation techniques, which are consistent and robust to various forms of serial correlation.

You now decide to run another model with the goal of examining how wages have fluctuated during this period.

We run the model from Part G and obtain the following results:

Why might we be concerned about the possibility of serial correlation given the model as specified in parts G and H?

We might be concerned about the possibility of serial correlation given the model as specified in parts G and H because the coefficients may be biased and the standard errors may be too small, leading to unreliable hypothesis tests. This may occur if the error term in the model is serially correlated, which violates the assumption of independent error terms.

You believe that your model from part H has an endogeneity problem. What changes can you make to your estimation to correct for any potential serial correlation?

To correct for the endogeneity problem in our model from part H, we can use instrumental variables (IV) estimation techniques, which involve selecting an instrument that is correlated with the endogenous variable but uncorrelated with the error term. This approach can help to eliminate the bias caused by endogeneity and improve the reliability of our estimates.

Suppose that you are worried that your model from part H might have a heteroskedasticity problem. What are the two forms of heteroskedascitiy we might be worried about in a time series regression, and how can we test for their presence in our model above?

The two forms of heteroskedascitiy we might be worried about in a time series regression are conditional heteroskedasticity, which means that the variance of the error term varies over time, and structural heteroskedasticity, which means that the variance of the error term varies with the level of the independent variables. We can test for their presence using various diagnostic tests, such as White’s test, Park’s test, or the ARCH-LM test, and make adjustments to our model using robust standard errors or generalized autoregressive conditional heteroskedasticity (GARCH) models.

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