The Simple Linear Regression Model

The Simple Linear Regression ModelOrdinary Least Square MethodSimple linear regression modele.g : Supply FunctionY = f(X)Where Y = quantity supplied X = price of the commodity

Assuming that variables are related with the simplest possible mathematical form, so the supply function in linear form isYi = b0 +b1Xib0 and b1 are the parameters of supply function and should be estimated its numerical value, o and 1

o should be either zero or positive ( o 0 )

In the particular case of supply function, the sign of 1 expected to be positif ( 1 > 0)

It is important to examine the relationship between the price elasticity of supply and the coefficient , 0 and 1

Elasticity from a regression line, use estimated and the mean value of price ( ) and quantity ( )

Remember: so, the form would be

Given that , it follows that:The supply will be elastic ( ) if is negative ( )The supply will be inelastic ( ) if is positive ( )The supply with have unitary elastic if

The above form implies that the relationship between X and Y is exact, there is no other factors that is effecting Y.But, when we gather the observation about the quantity supplied in the market with various price, well have the diagram e. g :

The deviation of the observation from line may be attributed to several factorsOmission of variables from the functionRandom behaviour of the human beingImperfect spesification of the mathematical from the modelErrors of aggregationError of measurement

Error in econometic function is usually donated by the letter u and is called error term or random disturbance term or stochastic term

The function model would be

The true relationship which connects the vaiables involved is splite into two parts : A part represented by a line A part represented by the random term u

Look the figure(figure shows the term d refers to term u)

= + = +

Systematic VariationVariation in YRandom VariationUnexplainedVariationExplained VariationVariation in Y

To estimate the coefficient b0 and b1 we need observation on X, Y and u. Yet, u is never like other explanatory variables, and therefore in order to estimate the function

we should guess the values of u, that is we should make some reasonable assumptions of each ui (its mean, variance and covariance)

Assumption of the linear stochastic regression modelStochastic assumption of ordinary lest squareAssumption 1 (randomness of u)ui is a random real variable : it may bepositif, negative or zeroAssumption 2 (zero mean of u)The mean value of u in any particular period is zero

Assumption 3 (homoscedasticity)The variance of ui is constant in each period

Assumption 4the variable ui has a normal distribution

Normal Distribution

Assumption 5 (Nonautocorrelation)The random term of different observation (ui, uj) are independent: covariance of any ui with any other uj are equal to zerofor

Assumption 6u is independent of explanatory variable(s) : their covariance is zero

Assumption 6AThe Xis are set of fixed value in the hypothetical process of repeated sampling which underlies the linear regression modelAssumption 7 (No errors of measurement in the Xs)The explanatory variable(s) are measured without error

Other assumption of ordinary least square

Assumption 8 (Noperfect multicolinear Xs)The explanatory variables are not perfectly linearly correlatedAssumption 9The macrovariables should be correctly aggregatedAssumption 10The relationship being estimated is identifiedAssumption 11The relationship is correctly specifiedThe distribution of the dependent variable YDependent variable Y has a normal distribution with mean

And variance

Proof 1.Given

Taking expected values we find

Bu using assumption 6

Furthermore, by assumption 2

Therefore,

Proof 2.

By assumption 3, the uis are homoscedastic, that is, they have the constant variance

Therefore,

The least square criterion and the normal equation of OLSThe true relationship between X and Y is

The true regression line is

And the estimated relationship is

And the estimated regression line is

Where= estimated value of Y, given a specified value of X=estimate of the true intercept= estimate the true parameter= estimate of th true value of the random term u

the striped line shows estimated regression lineAnd the light line shows the true regression line

Clearly deviation of the observation from the lines depend on their constant intercept ( ) and their slope ( ). The choice among all possible lines is done on the basis of what is called the least squares criterion.the least squate method should now be clear: the method seeks the minimisation of the sum of the squares of the deviation of the actual observation from the line

To estimate the function by calculating the and value, we can use these form

Or by using the deviation of the variables from the data mean

Worksheet for the estimation supply nYiXiXi2XiYiyixixiyixi2169981621600027612144912133399352636312-11-333945610100560-71-71557981513-600067710100770141141758749406-5-2104855864440-8-18196712144804431291053636318-10-33091172111217929218412648645121-1-11Estimation of a function whose intercept is zeroe.g : linear production function of manufactured products should normally have zero interceptEstimated function

Imposing the restrictionTherefore,

Estimation of elasticities from an estimated regression lineEstimated function

The derivative of with respect to X

If the estimated function is a linear demand of supply function the coefficient is not the price elasticity, but a component of the elasticity, which defined by

Where = price elasticityY = quantityX = price

Clearly is the componentFrom the estimated function we obtain an average elasticity

Where= the average price = average regressed value of the quantity, i.e the mean value of the estimated fromthe regression s= average value of the quantity

Note that , that is,the mean of the estimated value of Y is equal to the mean of the actual (sample) values of Y, because