## Questions and Answers

What is critical to the valid interpretation of the regression estimates?

The assumptions made about the X variable(s) and the error term

How many assumptions does the classical linear regression model (CLRM) make?

7

What is the first assumption of the CLRM?

Linearity

What does linearity in the CLRM assumption 1 refer to?

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What is the second assumption of the CLRM?

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What is meant by 'fixed X values' in the CLRM assumption 2?

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Why do we assume that the X values are non-stochastic?

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What is an example of a fixed X value in the context of consumption and income?

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What is the purpose of assuming no outliers in the X values in the CLRM?

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What are the key aspects to consider when specifying an econometric model?

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What is the consequence of choosing the wrong variables or functional form in an econometric model?

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Why is it important to ensure the regression model is correctly specified?

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What might happen if there are outliers in the X values in the CLRM?

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What is Assumption 9 of the CLRM?

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What is the name of the model when the X variable(s) is stochastic?

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What is the assumption about the mean value of the random disturbance term 𝒖𝒊 in CLRM?

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What do the distances above and below the mean values in the PRF represent?

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What is the implication of the assumption E(𝑢$ /𝑋$ ) = 0?

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What does Assumption 3 imply about the regression model?

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What is the relationship between E(𝑢$ /𝑋$ ) = 0 and E(𝑌$ | 𝑋$ ) = 𝛽% + 𝛽& 𝑋$?

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What is the difference between the stochastic regressor model and the fixed regressor model?

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What is the purpose of Assumption 3 in CLRM?

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What is the assumption that postulates the disturbances 𝑢ₗ and 𝑢ₑ are uncorrelated?

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What type of correlation is exhibited in Figure (a)?

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What is the characteristic of the disturbances in Figure (c)?

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What is the consequence of having disturbances that follow systematic patterns, such as those shown in Figures (a) and (b)?

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Why is Assumption 5 important in linear regression analysis?

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What is the implication of having positively correlated disturbances, 𝑢ₗ and 𝑢ₑ?

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What does the lack of autocorrelation between the disturbances imply?

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What is the intuitive explanation of Assumption 5?

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What type of error occurs when important explanatory variables are left out or unnecessary variables are included?

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What is the implication of Assumption 3 in CLRM?

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Why is it essential to assume that X and u are uncorrelated?

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What happens if X and u are positively correlated?

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What is another way of stating Assumption 3?

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What is CLRM Assumption 4?

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What is the conditional variance of 𝑢?

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What is the equation that represents the variance of 𝑢$ for each 𝑋$?

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## Study Notes

### Classical Linear Regression Model (CLRM) Assumptions

- The CLRM makes 7 assumptions that are critical to the valid interpretation of regression estimates.

### Assumption 1 - Linearity

- The regression model is linear in parameters, not necessarily in variables.
- The model can be extended to multivariate models.
- Linearity is in parameters, not variables.

### Assumption 2 - Fixed X Values

- Fixed X values or X values independent of the error term (strict exogeneity).
- Values taken by the regressor X may be considered fixed in repeated samples or sampled along with the dependent variable Y.
- In the latter case, it is assumed that the X variable(s) and the error term are independent: cov(X, u) = 0.

### Assumption 2 - Why assume fixed Xs?

- Consider the example of consumption (Y) vs income (X) in lecture 3.
- Keeping the value of income X fixed, we can randomly draw the following family consumption.
- The value of X is fixed at $80, and the process can be repeated for all the X values.
- If the X variable(s) is stochastic, the resulting model is called the neo-classical linear regression model (NLRM), in contrast to the CLRM, where the X's are treated as fixed or nonrandom.

### Assumption 3 - Zero Mean of the Disturbance

- Zero Mean Value of Disturbance ui: Given the value of Xi, the mean, or expected, value of the random disturbance term ui is zero.
- Symbolically, E(ui | Xi) = 0 or E(ui) = 0.
- The assumption implies that the factors not explicitly included in the model, and therefore subsumed in u, do not systematically affect the mean value of Y.
- In other words, the positive ui values cancel out the negative ui values so that their average or mean effect on Y is zero.
- E(ui | Xi) = 0 implies that E(Y | Xi) = β0 + β1Xi.

### Assumption 4 - Homoscedasticity or Constant of the Variance of the Disturbance

- The variance of ui for each Xi (i.e., the conditional variance of ui) is some positive constant number equal to σ^2.

### Assumption 5 - Non Autocorrelation between the Disturbances

- Equation 3.2.5 postulates that the disturbances ui and uj are uncorrelated.
- Technically, this is the assumption of no serial correlation, or no autocorrelation.
- This means that, given Xi, the deviations of any two Y values from their mean value do not exhibit patterns such as those shown in Figures (a) and (b).

### Assumption 7 - Nature of the X Variables

- There can be no outliers in the values of the X variable, that is, values that are very large in relation to the rest of the data.
- The requirement that there are no outliers in the X values is to avoid the regression results being dominated by such outliers.

### Assumption 9 - The Regression Model is Correctly Specified

- There is no specification bias or error in the model used in the empirical analysis.
- As discussed earlier, an econometric investigation begins with the specification of the econometric model, including what variables should be included, what is the functional form of the model, and what probabilistic assumptions are made about Y, X, and u.
- It will be shown later that the choice of wrong variables, wrong functional form, etc. will lead to questionable estimations.

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## Description

The classical linear regression model makes 7 assumptions which are critical to the valid interpretation of the regression estimates. This quiz covers these assumptions in the context of the two-variable regression model.