CLRM – Assumption 1: Linear Parameter and correct model specification

Assumption 1 requires that the dependent variable \textbf{y} is a linear combination of the explanatory variables \textbf{X} and the error terms \epsilon. Assumption 1 requires the specified model to be linear in parameters, but it does not require the model to be linear in variables. Equation 1 and 2 depict a model which is both, linear in parameter and variables. Note that Equation 1 and 2 show the same model in different notation.

(1) \textbf{y} = \textbf{X}\boldsymbol\beta + \boldsymbol\epsilon

(2) y_{i} = \beta_{0} + \beta_{1}x_{i1} + \beta_{2}x_{i2} + ... + \beta_{K}x_{iK} + \epsilon_{i}

In order for OLS to work the specified model has to be linear in parameters. Note that if the true relationship between x_{1} and latex y is non linear it is not possible to estimate the coefficient \beta in any meaningful way. Equation 3 shows an empirical model in which \beta_{1} is of quadratic nature.

(3) y_{i} = \beta_{0} + (\beta_{1})^{2}x_{i1} + \beta_{2}x_{i2} + ... + \beta_{K}x_{iK} + \epsilon_{i}

Assumption 1 of CLRM requires the model to be linear in parameters. OLS is not able to estimate Equation 3 in any meaningful way. However, assumption 1 does not require the model to be linear in variables. OLS will produce a meaningful estimation of \beta_{1} in Equation 4.

(4) y_{i} = \beta_{0} + \beta_{1}(x_{i1})^{2} + \beta_{2}x_{i2} + ... + \beta_{K}x_{iK} + \epsilon_{i}

Using the method of ordinary least squares (OLS) allows us to estimate models which are linear in parameters, even if the model is non linear in variables. On the contrary it is not possible to estimate models which are non linear in parameters, even if they are linear in variables.

Finally, every model estimated with OLS should contains all relevant explanatory variables and all included explanatory variables should be relevant.

 

Assumptions of Classical Linear Regressionmodels (CLRM)

Overview of all CLRM Assumptions
Assumption 1
Assumption 2
Assumption 3
Assumption 4
Assumption 5

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8 Responses to CLRM – Assumption 1: Linear Parameter and correct model specification

  1. Pingback: Assumptions of Classical Linerar Regressionmodels (CLRM) | Economic Theory Blog

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  7. Jane says:

    I have a question with regards to this. If for instance, the model if: ln(y) = a+bx^2 +e. do we still say that it satisfied GMA?

    • ad says:

      Generally, the Assumptions of Classical Linear Regression Model require the specified model to be linear in parameters, but they do not require the model to be linear in variables. Does this answer your question?

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