**CNLRM (Classical Normal Linear Regression Model)**

**Assumption**** 1: Linear regression model:** The regression model is linear in the parameters, as shown as

*Y**i *= β1 + β2*X**i *+ *u**i*

**Assumption 2: ***X *values are fixed in repeated sampling. Values taken by the regressor *X *are considered fixed in repeated samples. More technically, *X *is assumed to be *non- stochastic.*

**Assumption 3: Zero mean value of disturbance ***u**i***. **Given the value of *X*, the mean, or expected, value of the random disturbance term *u**i *is zero. Technically, the conditional mean value of *u**i *is zero. Symbolically, we have

E(ui |Xi) = 0 .

**Assumption 4: Homoscedasticity or equal variance of ***u**i***. **Given the value of *X*, the variance of *u**i *is the same for all observations. That is, the conditional variances of *u**i *are identical. Symbolically, we have

var (*u**i *|*X**i*) = *E*[*u**i *− *E*(*u**i *|*X**i*)]2

= *E*(*u**i*2 |*X**i *) because of Assumption 3

** **= σ2

where var stands for variance.

**Assumption 5: No autocorrelation between the disturbances. **Given any two *X *values, *X**i *and *X**j *(*i * *j *), the correlation between any two *u**i *and *u**j *(*i ≠ j *) is zero. Symbolically,

cov(*u**i**,u**j *|*X**i**,X**j*) = *E*{[*u**i *−*E*(*u**i*)]|*X**i*}{[*u**j *−*E*(*u**j*)]|*X**j*}

= *E*(*u**i *| *X**i*)(*u**j *| *X**j*)

=0

where *i *and *j *are two different observations and where **cov **means **covariance. **

**Assumption 6: Zero covariance between ***u**i *and *X**i*, or *E*(*u**i**X**i***) = 0.** Formally,

cov (*u**i*, *X**i*) = *E *[*u**i *− *E*(*u**i*)][*X**i *− *E*(*X**i*)]

= *E *[*u**i *(*X**i *− *E*(*X**i*))] since *E*(*u**i*) = 0

= *E*(*u**i**X**i*) − *E*(*X**i*)*E*(*u**i*) since *E*(*X**i*) is non-stochastic

= *E*(*u**i**X**i*) since *E*(*u**i*) = 0

= 0 by assumption

**Assumption 7: The number of observations ***n *must be greater than the number of parameters to be estimated. Alternatively, the number of observations *n *must be greater than the number of explanatory variables.

**Assumption 8: Variability in ***X *values. The *X *values in a given sample must not all be the same. Technically, var (*X *) must be a finite positive number.

**Assumption 9: The regression model is correctly specified. **Alternatively, there is no **specification bias or error **in the model used in empirical analysis.

**Assumption 10: There is no perfect multicollinearity. **That is, there are *no perfect linear relationships among the explanatory variables. *

Classical Linear Regression Model AssumptionsCLRM Assumptions
## Classical Linear Regression Model [CLRM/CNLRM]

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CNLRM (Classical Normal Linear Regression Model)Assumption1: Linear regression model:The regression model is linear in the parameters, as shown asYi= β1 + β2Xi+uiAssumption 2:Values taken by the regressorXvalues are fixed in repeated sampling.Xare considered fixed in repeated samples. More technically,Xis assumed to benon- stochastic.Assumption 3: Zero mean value of disturbanceui.Given the value ofX, the mean, or expected, value of the random disturbance termuiis zero. Technically, the conditional mean value ofuiis zero. Symbolically, we haveE(ui |Xi) = 0 .

Assumption 4: Homoscedasticity or equal variance ofui.Given the value ofX, the variance ofuiis the same for all observations. That is, the conditional variances ofuiare identical. Symbolically, we havevar (

ui|Xi) =E[ui−E(ui|Xi)]2=

E(ui2 |Xi) because of Assumption 3where var stands for variance.

Assumption 5: No autocorrelation between the disturbances.Given any twoXvalues,XiandXj(ij), the correlation between any twouianduj(i ≠ j) is zero. Symbolically,cov(

ui,uj|Xi,Xj) =E{[ui−E(ui)]|Xi}{[uj−E(uj)]|Xj}=

E(ui|Xi)(uj|Xj)=0

where

iandjare two different observations and wherecovmeanscovariance.Assumption 6: Zero covariance betweenuiandXi, orE(uiXi) = 0.Formally,cov (

ui,Xi) =E[ui−E(ui)][Xi−E(Xi)]=

E[ui(Xi−E(Xi))] sinceE(ui) = 0=

E(uiXi) −E(Xi)E(ui) sinceE(Xi) is non-stochastic=

E(uiXi) sinceE(ui) = 0= 0 by assumption

Assumption 7: The number of observationsAlternatively, the number of observationsnmust be greater than the number of parameters to be estimated.nmust be greater than the number of explanatory variables.Assumption 8: Variability inTheXvalues.Xvalues in a given sample must not all be the same. Technically, var (X) must be a finite positive number.Assumption 9: The regression model is correctly specified.Alternatively, there is nospecification bias or errorin the model used in empirical analysis.Assumption 10: There is no perfect multicollinearity.That is, there areno perfect linear relationships among the explanatory variables.Classical Linear Regression Model AssumptionsCLRM Assumptions