Classical Linear Regression Model Assumptions

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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?

What is the first assumption of the CLRM?

Linearity

What does linearity in the CLRM assumption 1 refer to?

Linearity in parameters, not variables Signup and view all the answers

What is the second assumption of the CLRM?

Fixed X values or X values independent of the error term Signup and view all the answers

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

X values that can be considered fixed in repeated samples Signup and view all the answers

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

To ensure that the X values are independent of the error term Signup and view all the answers

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

Income level of $80 Signup and view all the answers

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

To avoid the regression results being dominated by such outliers Signup and view all the answers

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

Variables to include, functional form of the model, and probabilistic assumptions about Y, X, and u Signup and view all the answers

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

Questionable estimations Signup and view all the answers

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

To avoid specification bias or error in the model Signup and view all the answers

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

The regression results may be dominated by such outliers Signup and view all the answers

What is Assumption 9 of the CLRM?

The Regression Model is Correctly Specified Signup and view all the answers

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

Neo-classical linear regression model (NLRM) Signup and view all the answers

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

The mean value of the random disturbance term 𝒖𝒊 is zero. Signup and view all the answers

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

The distances above and below the mean values represent the 𝒖𝒊. Signup and view all the answers

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

The positive 𝑢$ values cancel out the negative 𝑢$ values, resulting in a mean effect of zero on Y. Signup and view all the answers

What does Assumption 3 imply about the regression model?

The regression model is correctly specified. Signup and view all the answers

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

The two assumptions are equivalent. Signup and view all the answers

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

The stochastic regressor model treats the X's as stochastic, while the fixed regressor model treats the X's as fixed or nonrandom. Signup and view all the answers

What is the purpose of Assumption 3 in CLRM?

To ensure that the factors not explicitly included in the model do not systematically affect the mean value of Y. Signup and view all the answers

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

Assumption 5 – Non Autocorrelation between the Disturbances Signup and view all the answers

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

Positive correlation Signup and view all the answers

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

No systematic pattern or zero correlation Signup and view all the answers

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

Auto- or serial correlation Signup and view all the answers

Why is Assumption 5 important in linear regression analysis?

It ensures that the errors are not systematically related to each other Signup and view all the answers

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

𝑌ₗ depends not only on 𝑋ₗ but also on 𝑢ₑ Signup and view all the answers

What does the lack of autocorrelation between the disturbances imply?

The disturbances do not exhibit patterns such as those shown in Figures (a) and (b) Signup and view all the answers

What is the intuitive explanation of Assumption 5?

The disturbances should not be systematically related to each other Signup and view all the answers

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

Specification error Signup and view all the answers

What is the implication of Assumption 3 in CLRM?

𝑿𝒊 and 𝒖𝒊 are uncorrelated Signup and view all the answers

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

To assess their individual effects on Y Signup and view all the answers

What happens if X and u are positively correlated?

X increases when u increases and decreases when u decreases Signup and view all the answers

What is another way of stating Assumption 3?

There is no specification error in the chosen regression model Signup and view all the answers

What is CLRM Assumption 4?

Homoscedasticity or Constant Variance of the Disturbance Signup and view all the answers

What is the conditional variance of 𝑢?

Some positive constant number equal to 𝜎 Signup and view all the answers

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

Eq. 3.2.2 Signup and view all the answers

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.