Regression analysis is commonly used in various fields, such as economics, finance, engineering, social sciences, and healthcare, to model the relationships between variables and make predictions or forecast future outcomes.
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In which fields is Regression analysis commonly used?
What does inference for regression involve?
Inference for regression involves testing hypotheses, estimating the magnitude and direction of the relationship, and assessing the fit and predictive power of the regression model. This technique enables researchers and practitioners to make informed decisions based on statistical evidence and draw valid conclusions about the problem being studied.
If our aim is both estimation and inference, that is, if we aim to conduct hypothesis testing and construct confidence intervals for the parameters, what do we need to know?
If our aim is both estimation and inference, that is, if we aim to conduct hypothesis testing and construct confidence intervals for the parameters, we need to know the distribution of the estimators βˆ and s2.
What is the difference between standard normal distribution and t-distribution?
The standard normal distribution and the t-distribution are probability distributions used in statistics. The t-distribution is used when the population parameter is unknown and the sample is small, while the standard normal distribution is used when the population parameters are known or large sample sizes are available.
Which test is used for testing a hypothesis about any individual partial regression coefficient?
We use the t test for testing a hypothesis about any individual partial regression coefficient.
In order to make hypothesis tests, under both classical linear assumptions and normality we need t-distribution with n – k – 1 degrees of freedom. What is the formula for calculating the t-statistic?
where k + 1 is the number of unknown parameters in the population regression model. This is a crucial tool that enables us to test hypotheses related to the βj coefficients.
The population model is
In practice, we are primarily interested in testing the null hypothesis 
where j represents any of the independent variables. What does this null hypothesis imply?
The null hypothesis implies that “Xj has no effect on Y”
How should the alternative hypothesis be chosen?
The alternative hypothesis should be chosen based on the research question or the hypothesis being tested. Typically, it is formulated as the opposite of the null hypothesis, such as “βj ≠ 0” (two sided) or “βj < 0” or “βj > 0” (one sided), depending on the specific hypothesis being tested for the regression coefficient.
What are the steps for conducting a hypothesis test?
The general procedure for conducting a hypothesis test includes the following steps:
- Formulate the null hypothesis (H0) and the alternative hypothesis (H1) based on the research question or problem being investigated.
- Select a significance level (α) that represents the maximum probability of making a Type I error. Common values for α include 0.05 and 0.01.
- iii. Compute the value of the t-test statistic using the sample data, tβˆj.
- Determine the critical value (CV) at the chosen level of significance and the sample size. For one-sided test CV is tα.n–k–1, for two-sided test CV is tα/2.n–k–1
- Compare the computed t-value with the critical value.
- Interpret the results in the context of the research question or problem being investigated.
Using the data Determinants of House Prices gives the estimated equation:
where standard errors are given in parentheses below the estimated coefficients. We want to test whether the size of a house has a positive effect on the house price after controlling its age. Therefore, our null and alternative hypotheses are H0 : βj = 0 versus H1 : βj > 0. What is the value of t statistics?
What is the decision rule for hypothesis testing with the confidence-interval approach?
If the null hypothesis value is outside the confidence interval, then we reject the null hypothesis and conclude that the alternative hypothesis is supported. If the confidence interval includes null hypothesis value, then we fail to reject the null hypothesis.
Using the house price data, we estimate the following equation
where standard errors are given in parentheses. What is the confidence interval for the elasticity of price with respect to square footage?
To construct the confidence interval for the elasticity, we can use the coefficient estimate and its standard error. In our example, the coefficient estimate for the logarithm of square footage is 0.580 and its standard error is 0.101. For 95% confidence interval for the price elasticity we also need critical value with n – k – 1 = 10 – 2 – 1 = 7 degrees of freedom, t0.025,7 = 2.365. Using (4.10), we calculate the lower bound as 0.580 – 2.365 * 0.101 = 0.341 and upper bound as 0.580 + 2.365 * 0.101= 0.818. The confidence interval for the elasticity is approximately 0.341 to 0.818. This means we can be 95% confident that the true elasticity of price with respect to square footage falls within this range.
House price elasticity equation is estimated as follows:
Suppose we want to test whether the combined effect of the size and age of a house on the house price is significant or not. What is the null hypothesis?
The null hypothesis is:
What is the formula for calculating
?
What is the F Test used for?
It is used for testing the overall significance of a group of coefficients or the significance of a specific linear combination of the coefficients in a regression model.
The ANOVA approach partitions the total variation in the dependent variable into two components. What are these 2 components?
The explained variation (due to the regression model) and the unexplained variation (residuals or errors).
For 3-variable regression model;
In the ANOVA table for k-variable regression model, what is the degrees of freedom (df) for Residual Sum of Squares (RSS) ?
The degrees of freedom (df) for Residual Sum of Squares (RSS) is n-k.
Consider a regression model with the dependent variable being ‘sales’, the independent variables being ‘advertising expenditure’ and ‘price’. Let’s assume we have a sample size of 100 and we can construct the ANOVA table:
What is the value of F statistics?
F statistics is: 193.85
What is the relationship between R2 and F?
There is a positive relationship between R-squared and the F-statistic. A higher R-squared value indicates a better fit of the regression model, which is typically associated with a larger F-statistic and a more significant overall model.
Suppose that the unrestricted house price equation includes 4 independent variables: size, age, location (indicating the location of the house in a city or countryside) and interest rate. The estimated unrestricted model is:
Suppose that in the restricted model, we exclude location and interest rate. The estimated restricted model is:
Here, to test whether the location of the house and interest rate have no effect on house price, and the null hypothesis is
H0 : β3 = 0, β4 = 0 What is the value of F statistics for testing this hypothesis?
F-ratio or F-statistic is
