Regression Analysis Quiz

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Questions and Answers

What does SSE represent in regression analysis?

  • Regression sum of squares
  • Sum of squares due to error (correct)
  • Explained sum of squares
  • Sum of squares total

If a regression model does not include an intercept, then SST equals SSR plus SSE.

False (B)

What is the relationship of R² to SSE and SST?

R² = 1 - SSE / SST

In the equation R² = SSR / SST, if R² is equal to 1, then SSE is equal to _____ .

<p>0</p> Signup and view all the answers

Match the terms with their definitions:

<p>SST = Total variation in y SSR = Variation explained by the regression SSE = Variation not explained by the regression R² = Proportion of variance explained by the model</p> Signup and view all the answers

When the R² value is between 0 and 1, what does it represent?

<p>The proportion of variation in y explained by the model (D)</p> Signup and view all the answers

The correlation coefficient ρxy is calculated using the covariance of x and y divided by the product of their standard deviations.

<p>True (A)</p> Signup and view all the answers

What does the parameter β₂ represent in a log-linear model?

<p>The percentage change in y for a one unit increase in x.</p> Signup and view all the answers

The formula for the sample correlation coefficient rxy is rxy = ____ / (σx σy).

<p>ôxy</p> Signup and view all the answers

Match the following functional forms with their descriptions:

<p>Polynomial = Involves raising the variable x to a power Natural logarithm = Transformation using ln(x) Reciprocal = Involves using 1/x Log-log model = Both dependent and independent variables are transformed using logarithms</p> Signup and view all the answers

Which of the following transformations helps interpret elasticity?

<p>Log-log model (B)</p> Signup and view all the answers

Changing the scale of y affects the t-ratio but not the R².

<p>False (B)</p> Signup and view all the answers

What does it mean when R² equals r² in the context of multiple regressions?

<p>It shows the closeness between observations and their predicted values.</p> Signup and view all the answers

Which of the following components is included in the formula for the variance of the forecast error?

<p>Model uncertainty (B)</p> Signup and view all the answers

The standard error is calculated as the square of the variance of the forecast error.

<p>False (B)</p> Signup and view all the answers

What does the symbol $ŷ_0$ represent in the context of least squares prediction?

<p>$ŷ_0$ represents the predicted value of $y$ for a given value $x_0$.</p> Signup and view all the answers

The total sum of squares (SST) measures the total variation in $y$ about the sample mean, while the sum of squares due to the regression (SSR) reflects the variation ___.

<p>explained by the regression</p> Signup and view all the answers

What does the variance of the forecast error depend upon?

<p>Sample size and the variance of the regressor (B)</p> Signup and view all the answers

When calculating the prediction interval, the value $t_{n-2}$ is used to account for variability in predicted values.

<p>True (A)</p> Signup and view all the answers

Define the forecast error in the context of least squares prediction.

<p>The forecast error is the difference between the actual value $y_0$ and the predicted value $ŷ_0$.</p> Signup and view all the answers

Match the following terms with their definitions:

<p>SST = Total variation in y about the sample mean SSR = Variation explained by the regression model se(f) = Standard error of the forecast error var(f) = Variance of the forecast error</p> Signup and view all the answers

What is a crucial aspect of the functional form in regression models?

<p>It must satisfy assumptions SLR1-SR6. (C)</p> Signup and view all the answers

Visual inspection of residuals is sufficient for model validation.

<p>False (B)</p> Signup and view all the answers

What test is used to check for normality in regression errors?

<p>Jarque-Bera test</p> Signup and view all the answers

If a variable y has a normal distribution, then w = e^______.

<p>y</p> Signup and view all the answers

Match the following statistical terms with their definitions:

<p>Skewness = A measure of the asymmetry of the probability distribution Kurtosis = A measure of the 'tailedness' of the distribution Normal Distribution = A probability distribution that is symmetric about the mean Log-normal Distribution = A distribution of a variable whose logarithm is normally distributed</p> Signup and view all the answers

In the context of a log-linear model, how can predictions of y be optimally derived?

<p>By transforming the predictions back from the log scale using properties of the log-normal distribution. (B)</p> Signup and view all the answers

The median of a log-normal distribution is e^[μ].

<p>True (A)</p> Signup and view all the answers

What value do you compare the Jarque-Bera test statistic against to test for normality?

<p>5.99</p> Signup and view all the answers

Flashcards

Sum of Squares Due to Error (SSE)

The portion of the total variation in the dependent variable (y) that is not explained by the regression line. It represents the unexplained variation.

Sum of Squares Decomposition

The total variation in the dependent variable (y) is decomposed into two components: the variation explained by the regression (SSR) and the variation not explained (SSE). This principle holds when the intercept is included in the model.

Coefficient of Determination (R²)

The coefficient of determination, denoted by R², measures the proportion of the total variation in the dependent variable that is explained by the regression line. It ranges from 0 to 1, with higher values indicating a better fit.

R² = 1: Perfect Fit

When R² equals 1, all the sample data points fall perfectly on the regression line, indicating a perfect fit. It means there's no unexplained variation.

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R² = 0: Uncorrelated Data

If the data for y and x are uncorrelated, meaning they don't show a linear relationship, the regression line is horizontal, resulting in no explained variation and R² equaling 0.

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Least Squares Prediction

An estimate made by using a model to predict the value of a variable given a specific value of its predictor variable, based on the observed data.

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Forecast Error (f)

The difference between the actual value of the variable and the predicted value obtained using the regression line.

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Variance of Forecast Error (var(f))

The variance of the forecast error, which measures the uncertainty in our prediction of a new value.

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Sum of Squares Due to Regression (SSR)

A measure of how much of the total variability in the dependent variable is explained by the independent variable. It is the sum of the squared differences between the predicted values and the mean of the dependent variable.

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Total Sum of Squares (SST)

A measure of the overall variability in the dependent variable. It is the sum of the squared differences between the actual values and the mean of the dependent variable.

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R-squared (R²)

The ratio of the sum of squares due to regression (SSR) to the total sum of squares (SST). It represents the proportion of variability in the dependent variable that is explained by the regression model.

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R²

The proportion of the variation in y about its mean that is explained by the regression model.

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Correlation coefficient (ρxy)

A measure of the linear association between two variables, ranging from -1 to 1.

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Sample correlation coefficient (rxy)

The estimated correlation coefficient from a sample.

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R² and its relation to correlation coefficient

Used to assess the goodness-of-fit of a regression model. It is the square of the correlation coefficient between the observed and predicted values.

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Changing the scale of x in a regression model

When the independent variable (x) is transformed by multiplying it by a constant (c), the slope coefficient in the regression model changes by the same factor, while the intercept and error term remain unchanged.

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Changing the scale of y in a regression model

The slope and intercept coefficients in the regression model are divided by the constant (c), while the error term is also divided.

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Choosing a Functional Form for Regression

The regression model is linear in terms of the parameters, but the relationship between x and y can be non-linear. This is achieved by transforming the variables.

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Guidelines for Choosing a Functional Form

A guideline for choosing a functional form: use a transformation that makes the relationship between x and y appear linear and minimizes the variance of the error term.

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Visual inspection of residuals

A statistical method used to check the validity of a regression model by examining the distribution and patterns of the residuals (the difference between the observed and predicted values).

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Jarque-Bera Test

A measure of the deviation of a distribution from a normal distribution, considering both skewness and kurtosis. It tests the normality of the residuals.

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Log-Normal distribution

A statistical distribution used to model variables whose logarithm is normally distributed. Useful for describing positive-valued variables with a skewed distribution, like income or prices.

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Log-linear model

A regression model where the natural logarithm of the dependent variable (y) is expressed as a linear function of the independent variables (x). This model accounts for non-linear relationships between y and x.

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Exponentiating the log-linear model prediction

The exponential of the predicted value from a log-linear model. It provides the prediction for the original untransformed dependent variable (y).

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Rate of return in a wage equation

A statistical measure that provides an estimate of the average rate of change in a variable over a given period. It describes the expected percentage change in the dependent variable (y) for a one-unit change in the independent variable (x).

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Non-linear relationship

When the relationship between a dependent variable (y) and an independent variable (x) cannot be accurately represented by a straight line. There is a non-linear pattern in the data.

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Functional form

A mathematical function used to model non-linear relationships between variables. Example: Log-linear model, polynomial model, exponential model.

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Study Notes

Least Squares Prediction

  • Predict the value of y for a hypothetical x0
  • Assume the simple linear regression (SLR) model holds (SLR1-SLR5)
  • y0 = β1 + β2*x0 + ε0 (1)
  • Expected value of y0 given x0: E[y0|x0] = β1 + β2*x0
  • Predicted value of y0: ŷ0 = b1 + b2*x0 (2)

Forecast Error

  • Define the forecast error: f = y0 - ŷ0 = (β1 + β2*x0 + ε0) - (b1 + b2*x0) (3)
  • The expected value of the forecast error is zero: E[f] = 0
  • ŷ0 is an unbiased predictor of y0

Variance of the Forecast Error

  • Variance of the forecast error: var(f) = σ2 * [1 + (1/N) + ((x0 - x̄)2/∑(xi - x̄)2)] (4)
  • Depends on
    • Model uncertainty (σ2)
    • Sample size (N)
    • Variance of the regressor (x̄)
    • Value of (x0 - x̄)2

Estimated Forecast Error Variance

  • var(f) = σ2 * [1 + (1/N) + ((x0 - x̄)2/∑(xi - x̄)2)]

Standard Error

  • Standard error of the forecast (se(f)) is the square root of the variance of the forecast error.

Prediction Interval

  • Prediction interval: ŷ0 ± t(n-2, α/2) * se(f)

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