Beteta - Problem Set Module 3

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Southern New Hampshire University *

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303

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Mathematics

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Apr 3, 2024

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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

Our scatterplot shows how wage growth compares with the unemployment rate. The relationship between the two isn't a straight line, indicating that the connection isn't just a simple increase or decrease. The pattern suggests that as unemployment changes, the effect on wage growth might increase initially and then decrease, or the opposite, forming more of a curve than a straight line. The scatterplot shows that the relationship between wage growth and unemployment is not perfectly linear. The data points do not align in a straight line, which suggests that a first order (linear) model might not be the best fit. Instead, the data points show a pattern that could be a curve, hinting that unemployment's impact on wage growth isn't constant as unemployment changes. Given this observation, a second order (quadratic) model might be more appropriate, which would allow for a curve that can bend upwards or downwards. This model can account for a more complex relationship where the effect of unemployment on wage growth could increase or decrease at different unemployment rates rather than changing at a constant rate, as a linear model would suggest. Reporting Results Report the results of the regression model. Address the following questions in your analysis:  General Form: y = β 0 + β 1 x + β 2 x 2  Prediction Equation: ^ y = ^ β 0 + ^ β 1 unemployment + ^ β 2 unemployment 2 3

 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

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