A_Gonzalez_Module_Three_Problem_Set
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Southern New Hampshire University *
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Apr 3, 2024
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MAT 303 Module Three Problem Set Report
Second Order Models
Alfonso Gonzalez
Alfonso.Gonzalez@snhu.edu
Southern New Hampshire University
1. Introduction
For this module we are doing an examination of a dataset primarily focused on economic metrics, such as wage growth, unemployment rates, and potentially Gross Domestic Product (GDP) expansion. Our objective is to discern the intricate relationships among these economic variables and potentially forecast wage growth under varying economic circumstances. This analysis holds significant importance for policymakers, economists, and businesses alike, aiding in strategic planning and decision-making.
We will review the dataset labeled economics.csv to discover correlations between wage growth, unemployment, and other pertinent economic indicators.
The insights obtained from this investigation could inform economists in shaping policy directives, assist businesses in devising wage adjustment strategies, and empower job seekers to navigate prevailing market trends effectively.
Our analytical approach will involve conducting correlation analyses and employing regression models, with a specific emphasis on quadratic (second-order) models to accommodate potential nonlinear associations among variables.
2. Data Preparation
What are the important variables in this data set?
How many rows and columns are present in this data set? Once we load the dataset labeled economics, we can see that the dataset comprises 99 rows and 6 columns, indicating 99 observations across various economic indicators and conditions. The variables included in this dataset are:
wage_growth:
Percentage growth in wages.
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inflation:
Inflation rate.
unemployment
: Unemployment rate.
economy:
Economic condition, categorized as "recession" or "no_recession".
education:
Education level, with categories such as "college", "post_graduate", and "high_school".
gdp:
GDP growth rate.
The primary variables of interest for our analysis will be wage_growth and unemployment, as indicated by the problem set. Additional analyses will incorporate gdp and economy as predictor variables.
3. Quadratic (Second Order) Model with One Quantitative Variable
Correlation Analysis
To understand how wage growth and unemployment are connected, I'll make a graph called a scatterplot. This graph shows if there's a straight-line relationship between the two, or if it's more curved. Based on how the points on the graph look, we can decide if a simple straight line (linear model)
or a curve (quadratic model) better fits the data.
When I made the scatterplot comparing wage growth and unemployment, I noticed that the dots on the graph didn't form a perfect straight line. Instead, they seemed to follow a pattern that wasn't strictly going up or down in a straight line as unemployment changed. This makes me think that a curve might 3
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describe the relationship between wage growth and unemployment better than a straight line. So, using a quadratic model, which can handle curves in the data, might be a better fit than a simple straight line (linear) model.
Reporting Results Report the results of the regression model. Address the following questions in your analysis:
The general form of a second-order regression model with one quantitative variable is:
Where:
is the dependent variable (wage growth)
is the dependent variable (unemployment rate)
is the intercept of the model,
is the coefficient for the independent variable,
is the coefficient for the square of the independent variable, and
is the error term of the model.
Using the numbers we have; we can figure out how much wages might grow based on the unemployment rate.
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In this equation:
Wage Growth is the dependent variable we want to predict.
12.234206 is the estimated intercept.
-1.743170 is the coefficient for the linear term of Unemployment.
0.667408 is the coefficient for the linear term of Unemployment (Unemployment squared)
R-squared and Adjusted R-squared:
The value of for the model is 0.9436.
The Adjusted value is 0.9424.
The R-squared (R²) value for our model is 0.9436, meaning about 94.36% of the differences in wage growth can be explained by unemployment and its squared value. The Adjusted R-squared value, which considers the number of predictors in the model, is very close to the R-squared value. This suggests that our predictors are relevant, and there's no penalty for having unnecessary complexity in the model.
Interpretation of Beta Estimates:
The coefficient for unemployment (β₁ = -1.743170) shows that for every one percentage point increase in unemployment, wage growth tends to decrease by about 1.74 percentage points, when the squared term is kept constant. The positive coefficient for unemployment squared (β₂ = 0.667408) indicates a curved relationship between unemployment and wage growth. As unemployment goes up, the rate of decrease in wage growth slows down, especially when unemployment is higher. These coefficients suggest that the relationship between wage growth and unemployment isn't simple. Higher unemployment generally means lower wage growth, but the effect varies depending on how high the unemployment rate is.
Evaluating Model Significance Let's simplify the results from our regression analysis:
The second-order polynomial model we used is good at explaining how wage growth changes with the unemployment rate. The overall F-test, which looks at how well our model fits the data, gives us an F-
statistic of 803. With a p-value less than 2.2e-16, it tells us that our model with both unemployment and its square as predictors is way better at explaining wage growth than just using an intercept (no predictors). Both the linear term for unemployment and the quadratic term for unemployment squared are important. The p-values from the T-tests, which check the significance of each predictor, are super low—less than 2e-16 for unemployment and 6.07e-15 for unemployment squared. This means both terms have a big impact on predicting wage growth. In simple terms, our regression model is good at predicting wage growth. Both the linear and quadratic terms of unemployment are important for predicting wage growth, as shown by the low p-values in the T-tests.
Making Predictions Using Model 5
When the unemployment rate is 2.54, we predict that wage growth will be around 8.2414. This prediction considers both the direct impact of unemployment and its squared impact on wage growth.
Now, if we were to predict wage growth for different people or groups, each with an unemployment rate of 2.54, we'd expect the actual wage growth to land somewhere between 6.9071 and 9.5758 about 95% of the time. This range, called the 95% prediction interval, shows how uncertain our predictions are for individual cases, considering both errors in our average wage growth estimate and the natural differences in wage growth among people.
On the other hand, the 95% confidence interval for the average wage growth, assuming an unemployment rate of 2.54, is narrower. It ranges from about 8.0936 to 8.3893. Unlike the prediction interval, this range isn't about individual predictions but about the average wage growth. We can be 95% confident that the average wage growth for everyone with a 2.54 unemployment rate falls within this range. This interval is narrower because it only considers the uncertainty in estimating the mean, not the variability of individual data points.
4. Complete Second Order Model with Two Quantitative Variables
Reporting Results Report the results of the regression model. Address the following questions in your analysis:
Write the general form and the prediction equation of the complete second order regression model for wage growth as the response variable, and unemployment and GDP growth as predictor variables.
Create this second order regression model for wage growth as the response variable, and unemployment and GDP growth as predictor variables. Write the prediction model equation using outputs obtained from your R script. 6
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What are the values of (R-squared) and (Adjusted R-squared) for the model? Provide your interpretation of these statistics.
Interpret the beta estimates for GDP
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(GDP squared) and unemployment 2
(unemployment squared).
General Model:
This second-order model includes various factors that could affect wage growth. It looks like this:
Prediction Equation:
Based on our analysis, here’s the specific prediction equation for wage growth:
R-squared and Adjusted R-squared:
The R-squared (R²) value is 0.9587, which means about 95.87% of the changes in wage growth are explained by our model. The Adjusted R-squared value is 0.9565, which is a bit lower because it accounts for the number of predictors in the model. Still, it's high, showing our model fits the data well without being overly complex.
Interpretation of the Beta Estimates:
The coefficient for GDP squared (β5 = -0.066599) isn't statistically significant, suggesting the quadratic term for GDP doesn't really help predict wage growth much. On the other hand, the coefficient for unemployment squared (β4 = 0.837685) is significant, meaning the effect of unemployment on wage growth increases as unemployment goes up. This shows a non-linear relationship.
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In short, this model helps us understand how wage growth is influenced by unemployment and GDP growth. While unemployment squared matters, GDP squared doesn't seem to have much impact. The high R-squared and adjusted R-squared values indicate our model does a good job explaining wage growth variation, making it useful for understanding economic trends.
Evaluating Model Significance The importance of the regression model is figured out by checking two things: the overall F-test and the individual T-tests for each predictor.
First, for the overall model, we look at two possible scenarios: either none of the predictors help explain wage growth (null hypothesis), or at least one of them does (alternative hypothesis). The F-statistic is 432, and the p-value is less than 2.2e-16, much lower than the 0.05 threshold. So, we can say bye-bye to
the null hypothesis and conclude our model is significant at the 5% level, meaning all the predictors together help explain wage growth.
Now, when we zoom into each predictor, we find some interesting stuff. Unemployment, GDP growth, and the squared term for unemployment are all significant at the 5% level, with p-values below 0.05. However, the interaction between unemployment and GDP, and the squared term for GDP, don't reach this level of significance, with higher p-values.
To wrap it up, our model overall is super important, and specifically, unemployment, GDP growth, and the squared term for unemployment really help predict wage growth. But the interaction between unemployment and GDP, and the squared term for GDP, don't seem to add much at the typical 5% significance level.
Making Predictions Using Model The regression analysis has given us a forecast for wage growth based on an unemployment rate of 2.50 and a GDP growth rate of 6.50. According to the model, in this scenario, we'd expect wage growth to be around 7.806.
Now, let's talk about the 95% prediction interval for wage growth, which ranges from about 6.6315 to 9.8805. This interval tells us that if we took many samples from a population with the same 8
unemployment and GDP growth rates, 95% of the time, the actual wage growth would fall within this range. It's quite wide because it includes the uncertainty of predicting a single new data point and also considers how much wage growth might vary.
On the other hand, the 95% confidence interval for wage growth, ranging from approximately 7.7583 to 8.0289, gives us an estimate of where the average wage growth might lie for the whole population with these unemployment and GDP growth rates. This range is narrower because it only considers the uncertainty in figuring out the true average, not the variation of individual outcomes around that average.
So, in practical terms, if a policymaker or economist was using this model to predict wage growth or set economic policies, they could be 95% confident that the average wage growth would be between 7.7583 and 8.0289 with the specified levels of unemployment and GDP growth. But individual predictions could vary more, as shown by the prediction interval.
5. Complete Second Order Model with One Quantitative and One Qualitative Variable
Reporting Results The general form of the second order regression model is:
Using the provided coefficients, the specific prediction equation becomes:
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In this equation, the "Economy Recession" variable is like a switch where recession is labeled as 1, and no recession as 0. The term "Unemployment:Economy Recession" helps us see if unemployment affects wage growth differently during a recession or not.
The R-squared (R²) value of the model is 0.9453, showing that about 94.53% of the wage growth variation is explained by our model. The adjusted R-squared value is a bit lower at 0.9429, which considers the number of predictors compared to the number of observations. Even though it's slightly lower, it's still high, suggesting our model fits the data well and the predictors we've chosen are pretty good at explaining wage growth. This drop from R² to adjusted R² is expected, as the adjusted R² is a bit stricter, especially when we add more terms to the model.
Evaluating Model Significance Evaluate model significance for the regression model. Address the following questions in your analysis:
Is the model significant at a 5% level of significance? Carry out the overall F-test by identifying the null hypothesis, the alternative hypothesis, the P-value, and the conclusion of the test.
Which terms are significant in the model based on individual T-tests? Use a 5% level of significance.
The null hypothesis (H0) for the overall F-test suggests that none of the predictors really help explain the
changes in the dependent variable, wage growth. It's like saying we could get by with just the intercept. The alternative hypothesis (H1) says that at least one predictor does make a big difference.
With an F-statistic of 405.8 and a p-value way lower than the usual cutoff of 0.05 (actually, p < 2.2e-16), we can reject the null hypothesis. This means the model overall is important at the 5% level, meaning the predictors we've used help explain why wage growth changes.
Now, looking at each predictor on its own, the numbers tell us that both unemployment and its squared form are important. Their p-values are super tiny, meaning they make a big difference in the model. But when it comes to whether the economies in recession or not, and how unemployment acts during a recession, they don't seem to matter much. Their p-values are higher than 0.05, which means they're not so important at the 5% level.
So, to sum up, some predictors, like unemployment and its squared term, are useful for predicting wage growth, while others, like the state of the economy and how it interacts with unemployment, don't seem to matter much in this model.
Making Predictions Using Model 10
The model estimates that wage growth will be around 8.3396 when the unemployment rate is 2.50 and the economy is not in recession.
Now, if we predict wage growth for different people or groups under similar conditions—2.50 unemployment and no recession—we'd expect our predictions to fall between 7.0898 and 9.6695 about 95% of the time. This range shows how much wage growth can vary due to individual differences and the uncertainty in our prediction.
On the other hand, the narrower 95% confidence interval, from 8.1836 to 8.4957, gives us a better idea of where the average wage growth is for everyone under these conditions. It's not about the spread of individual predictions but how precise our average wage growth estimate is.
The prediction interval is wider than the confidence interval because it considers the extra variability introduced by individual data points. While the confidence interval just looks at uncertainty in estimating the true mean wage growth, the prediction interval also includes variability between individual wage growth figures, making it naturally broader.
6. Conclusion After analyzing the data, we found that our models are good at explaining wage growth. They shed light on the complicated connections between wage growth and things like unemployment and GDP. These models look at both straight-line relationships and more complicated ones, like when different factors interact.
I think these models should be used with caution, even though they're pretty solid. They show a strong link between the factors we studied and wage growth. But we need to be careful because not all the factors were important in every model. For example, in one model including GDP and unemployment, we found that the GDP squared term didn't really matter much. This might mean that the connection between GDP growth and wage growth is simple and doesn't need a complicated formula. Understanding this can help policymakers make better decisions about the economy.
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The importance of different factors, like the squared unemployment term, shows us how wage growth changes in different situations. This is useful for planning and predicting what might happen in the economy.
These analyses give us a good starting point for figuring out what might happen in the future. They can help with things like setting government policies, deciding on wages, and understanding the job market. But we should keep investigating and testing these models with more data to be sure they're reliable. It's crucial to make sure they match up with real-world economics and what we expect to see happening.
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