Beteta - Problem Set Module 3

MAT 303 Module Three Problem Set Report Second Order Models Diego Beteta diego.beteta@snhu.edu Southern New Hampshire University

1. Introduction The economic dataset includes variables such as wage growth, inflation, unemployment, economic conditions (recession or no recession), levels of education, and GDP. This data is likely a historical record intended to study how different economic factors are associated with wage growth in the labor force. Our results could be vital for policymakers and economists to understand the dynamics of wage growth, helping to inform decisions and create strategies for economic development. The analyses will likely involve statistical methods to determine correlations, trends, and potentially predictive modeling to estimate future wage growth under different economic scenarios. We might employ regression analysis to understand the relationships between wage growth and other factors and time-series analysis if the data is chronological to look at the trends over time. 2. Data Preparation The important variables in this dataset that we're focusing on include:  Wage Growth : This measures the percentage increase in labor force wages. Understanding this helps to evaluate workers' standard of living and economic prosperity.  Inflation : Inflation represents the rate at which the general level of prices for goods and services rises and, subsequently, how purchasing power is falling. Analyzing inflation alongside wage growth can indicate whether wage increases keep pace with the cost of living.  GDP (Gross Domestic Product) Growth : GDP growth is the increase in the production and consumption of goods and services in an economy. It's a broad measure of overall economic activity and health.  Unemployment: This is a measure of the number of people who are actively looking for work but are not currently employed. These variables are crucial as they interplay to define a country's or region's economic condition, influencing policy decisions. Regarding the structure of the dataset, it consists of 99 rows and 6 columns. Each row represents an entry (potentially a year or other time frame). In contrast, the columns represent the variables mentioned, including wage growth, inflation, GDP growth, and other related economic factors. 3. Quadratic (Second Order) Model with One Quantitative Variable Correlation Analysis 2

 Second-order regression model for wage growth using unemployment as the independent variable: ^ wage growth = 12.2342 − 1.7432 unemployment + 0.0674 unemployment 2  R-squared value = 0.9436 This value tells us that the model explains about 94.4% of the variance in wage growth. It's a measure of how well the observed outcomes are replicated by the model, based on the proportion of total variation of outcomes explained by the model.  Adjusted R-squared value = 0.9424 This value adjusts the R-squared for the number of predictors in the model and the number of observations. It's approximately 94.2%, close to the R-squared value. This similarity suggests that the number of predictors is appropriate for the number of observations in the model. Both statistics indicate that our second-order model does an excellent job of explaining how wage growth changes with unemployment. The high values mean that the model fits the data well, and the slight difference between R-squared and Adjusted R-squared implies that we are not penalized much for any extra complexity in the model; in other words, our model is appropriately complex given the data. The beta estimates from our second-order regression model for the terms related to unemployment are interpreted as follows:  Unemployment ( ^ β 1 = -1.7432) : This coefficient is negative, indicating an initial decrease in wage growth as unemployment increases. When unemployment rises by 1%, we expect wage growth to decrease by about 1.7432%, assuming we are in the range where the linear term dominates the relationship.  Unemployment Squared ( ^ β 2 = 0.0674) : This coefficient is positive, which tells us that there is a point where the effect of increasing unemployment starts to slow down the decrease in wage growth, or it might even begin to increase wage growth after a certain level. This term accounts for the curvature in the relationship. In practical terms, as unemployment continues to increase, the rate at which wage growth is falling (because of the negative linear term) starts to slow down. Eventually, the trend could reverse (meaning wage growth could increase after a certain point of unemployment). Together, these coefficients describe a relationship where, initially, unemployment increases are associated with wage growth decreases. Still, as unemployment rises, this effect slows down and could reverse due to the squared term. This quadratic relationship captures the more complex reality that the impact of unemployment on wage growth isn't constant and can change direction as unemployment levels change. Evaluating Model Significance The model is highly significant, with an F-statistic of 803 on 2 and 96 degrees of freedom and a p-value of less than 2.2e-16. This F-statistic is well above the critical value we would expect if the null hypothesis were true (typically around 3 or 4 for this sample size), and the p-value is far below the conventional 4

significance threshold of 0.05. These results lead us to confidently reject the null hypothesis, indicating a very strong relationship between unemployment (and its square) and wage growth within the data. This means that our model explains the variability in wage growth and that unemployment is a meaningful predictor in this context. Based on individual T-tests and using a 5% level of significance, both terms in the model— unemployment and unemployment squared—are significant. This significance is determined by looking at the p-values for each term's T-test, which are well below the 0.05 threshold. This means that unemployment's linear and quadratic components significantly impact wage growth. The data strongly suggests that as unemployment changes, it has a meaningful and statistically significant effect on how wages grow, both in the initial rate of change and the rate at which this change accelerates or decelerates. Making Predictions Using Model If the unemployment rate is 2.54, the predicted wage growth, according to our second-order regression model, is approximately 8.24%. This means that with an unemployment rate of 2.54%, we expect wages to grow by about 8.24% based on the historical data and the relationship we have modeled. The 95% prediction interval for wage growth when the unemployment rate is 2.54 is [6.91, 9.58]. This interval suggests that we can be 95% confident that the actual wage growth will fall between 6.91% and 9.58%, given the current model and the unemployment rate at 2.54. It accounts for the uncertainty in the prediction, acknowledging that there are other factors and random variations that our model does not capture. The 95% confidence interval for the predicted wage growth when the unemployment rate is 2.54 is [8.09, 8.39]. This confidence interval provides a range in which we are 95% confident that the true average wage growth would fall if we repeated our study many times under identical conditions. It's narrower than the prediction interval because it doesn't account for the individual variations that occur from one observation to the next but rather the uncertainty in estimating the true mean wage growth rate for the entire population at this level of unemployment. 4. Complete Second Order Model with Two Quantitative Variables Reporting Results  General Form: y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2 + β 4 x 1 2 + β 5 x 2 2  Prediction Equation: ^ y = ^ β 0 + ^ β 1 unemployment + ^ β 2 gdp + ^ β 3 unemployment 2 + ^ β 4 gdp 2 + ^ β 5 unemployment ∗ gdp  Second-order regression model for wage growth as the response variable and unemployment and GDP growth as predictor variables: ^ wage growth = 8.9894 − 1.1528 unemployment + 0.2837 gdp + 0.0377 unemployment 2 − 0.0066 gdp 2 − 0.0066 un 5

means our model effectively captures how wage growth is influenced by changes in unemployment and GDP growth, affirming the importance of these economic indicators in predicting wage trends. In the model, the individual terms are evaluated for significance using T-tests at a 5% level of significance. From the regression output, we can determine significance by looking at the p-values associated with each term's T-statistic. A term is considered significant if its p-value is less than 0.05. In our model, the terms with p-values below this threshold are unemployment, unemployment squared, and the intercept. These terms statistically affect wage growth and contribute meaningful information to the model's predictions. The coefficients for GDP and the interaction between unemployment and GDP were not statistically significant, indicating they don't add significant predictive value to the model within the range of data observed. Making Predictions Using Model If the unemployment rate is 2.50 and GDP growth is 6.50, according to our second-order regression model, the predicted wage growth is approximately 7.81%. This means that with an unemployment rate of 2.50% and GDP growth of 6.50%, we expect wages to grow by about 7.81% based on the historical data and the relationships we have modeled. The 95% prediction interval for wage growth when the unemployment rate is 2.50 and GDP growth is 6.50 is approximately [6.63, 9.00]. Given the current model and the specified unemployment and GDP growth levels, this interval suggests that we can be 95% confident that the actual wage growth will fall between 6.63% and 9.00%. It accounts for the uncertainty in the prediction, acknowledging that there are other factors and random variations that our model does not capture. The 95% confidence interval for wage growth [7.583, 8.0289] indicates that we can be 95% certain that the true average wage increases for the entire population, given an unemployment rate of 2.50% and GDP growth of 6.50%, would fall within this range. This confidence interval is not about the variability of individual predictions (like the prediction interval), but instead, it's about how precisely we have estimated the average wage growth rate. It suggests that, despite the individual variations and other factors at play, the average wage growth we expect under these economic conditions is quite narrowly defined, reflecting a high level of precision in our model's estimations. 5. Complete Second Order Model with One Quantitative and One Qualitative Variable Reporting Results Report the results of the regression model. Address the following questions in your analysis:  General Form: y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2 + β 4 x 1 2 + β 5 x 1 2 x 2  Prediction Equation: 7

^ y = ^ β 0 + ^ β 1 unemployment + ^ β 2 economy + ^ β 3 unemployment ∗ economy + ^ β 4 unemployment 2 + ^ β 5 unemployment 2 econo  The second-order regression model for wage growth using unemployment and economy as predictors is: ^ wage growth = 12.3607 − 1.8083 unemployment − 2.7040 economy + 0.0757 unemployment 2 + 0.6936 unemploy  R-squared value = 0.9475 This value indicates that the model explains approximately 94.75% of the variance in wage growth. It measures the goodness of fit, showing how well the observed outcomes are replicated by the model based on the proportion of total variation of outcomes explained.  Adjusted R-squared value = 0.9446 This value adjusts the R-squared for the number of predictors in the model and the number of observations. It is approximately 94.46%, close to the R-squared value. This proximity suggests that the number of predictors used is appropriate for the number of observations. Both statistics suggest that the second-order regression model is highly effective in explaining the changes in wage growth as influenced by unemployment and the state of the economy. The high R- squared values indicate a strong fit to the data, while the Adjusted R-squared shows that the model complexity is justified given the number of predictors. Evaluating Model Significance The regression model displays exceptional significance, with an F-statistic of 335.42 for incorporating the predictors, including their squared and interaction terms. The corresponding p-value is less than 2.2e- 16, which suggests an extremely low chance that such an F-statistic would arise if the null hypothesis were accurate, that is, if none of our predictors affected wage growth. Given that the F-statistic substantially surpasses the critical value expected if the null hypothesis were true and the p-value is well below the 0.05 threshold for significance, we can confidently reject the null hypothesis. This leads us to conclude that our model, which encompasses unemployment and the economic state (as well as their squared terms and interaction), maintains a potent and statistically significant correlation with wage growth. Practically, this implies that our model effectively captures the impact of changes in unemployment and economic conditions on wage growth, underscoring the predictive power of these economic indicators for wage trends. In the model, the significance of individual terms is assessed using T-tests at a 5% level of significance. The p-values associated with each term's T-statistic in the regression output inform us about their significance. A term with a p-value less than 0.05 is deemed significant. The terms in our model with p- values falling beneath this threshold include the intercept, unemployment, the square of unemployment, and the interaction between unemployment and the economic state. These terms have a statistically significant impact on wage growth and are important contributors to the model's predictive power. While included in the model, the coefficients for the economy recession and the interaction between squared unemployment and economy recession did not achieve statistical 8

guide decisions on economic policy, business strategy, and workforce planning. They allow us to anticipate how wages might change in response to unemployment and economic performance fluctuations, providing a basis for proactive rather than reactive decision-making. 10