Estimating the model
p = b1 + b2 ulc + ui
We use the classical linear regression model to estimate the above model. The assumptions of this model are that:
The error term is a random variable across observations
The mean value of the
error term is zero
The variance of the
error term is constant across observations
The random variable
assumes a normal distribution
The error terms are
The error term is uncorrelated with the explanatory variable
The model depict that the model is of the form:
Y = b1 + b2 X
Where Y is the dependent variable, b1 is the autonomous value, b2 is the slope of the model
and X is the independent variable. The model is therefore estimated as follows:
B1 = Y’- b2X’
Where Y’ is the mean of Y and X’ is the mean of X
B2= (∑ y x) / (∑ x2)
Where y is (Y-Y’), and x is (X –X’), which in other words is the observed variable minus the mean of the variable
After estimation using the above model the result of our model which is p = b1 + b2 ulc
p = 0.002437615 + 1.016405033 ulc
Interpretation of the estimated model:
As the model gives us that the autonomous value is equal to 0.002438 then this means that the value of p that is not affected by changes in the value of ulc is 0.002438, this means that if the value of ulc is zero then the value of p will be 0.002438. for the slope which is 1.0164, then if we hold all other factors constant and increase the value of ulc by one unit then the value of p will increase by 1.0164.
The value of the coefficient of determination R2 shows the strength of the relationship between
two variables, the value of R 2 rages from 0 to 1, when the level of R
is 1 then we have a strong relationship between the variables, if the value is near zero then we have a weak relationship between variables.
In our case we calculate the value of R2 using the following formula;
R2 = (b22 ∑x2 )/ ∑y2
After calculation the value is 0.99946344, this value is close to one, therefore we can conclude that we have a very strong and positive relationship between the two variables.
Hypothesis testing that labor costs has no effect on the cost of production
To test whether labor costs have an effect on the cost of production then we will have to test the statistical significance of the estimated parameters, this will involve testing hypothesis on both the autonomous value and the slope of the model we have estimated:
We now test the statistics by checking whether the T calculated value and the T critical value which will be checked on the t table, when the t calculated is greater than the t critical then we reject the null hypothesis which states that the slope is equal to zero, when we reject the null hypothesis then we are stating that the parameter under test is statistically significant.
When the T calculated is in the acceptance zone then we accept the null hypothesis that the parameter is equal to zero, if it is in the rejection zone then we reject the null hypothesis.
Our T calculated is = 251.6598206
Our T critical at 98% from the tables (two tail test) is equal to 2.45726,
Therefore our T calculated > T critical, we reject the null hypothesis that our slope is equal to zero. Therefore we reject the hypothesis that labor cost has no effect on the cost of production, from our test it is clear that labor cost have an effect on the cost of production.
Estimation of the model: p = b1 + b2 ulc + b2 umc
In this case we will still use the classical linear regression model. However we ill use the matrix form to solve this multiple regression model.
When a model takes the following form Y = b1 + b2X1 + b3X2 then the estimation by matrix will be done as follows
b = (x’x)-1(x’y)
Our resulting estimated parameters are as follows;
Therefore our model will be as follows
p = 0.001446633 + 0.980010361ulc + 0.031652364umc
From the above estimated model we can explain the autonomous value by stating that if we hold all other factors constant and that the value of ulc and umc is equal to zero then the value of p will be 0.0014466, if we hold all other factors constant and increase the value of ulc by one unit then the value of p will increase by 0.98 units, if also we all hold all other factors constant and increase the level of umc by one unit then the level of p will INCREASE by 0.03165 units.
B2 in pm1 and pm2
The value of our b2 in our estimated equation one is 1.016405033, in our estimated equation 2 the value of b2 is 0.98001036, the our first equation the b2 value is higher than the value in estimated equation two, the difference is because in the first equation we only have one independent variable while in equation two we have two independent variable, when we increase explanatory variables on an equation then the level at which one variable affects the dependent variable decreases.
In model estimation the R2 gives the strength of the relationship that exist between the explanatory variables and the dependent variable, for this reason there may be a strong relationship between the variables but the model estimated is not statistically significant. For this reason the r2 values compared may not give us a good comparison of the two estimated models. For this reason therefore the best way to test validity and the best model is by performing statistical test of hypothesis checking the statistical significance of the estimated variables in the model.
Hypothesis testing of b2 and b3
From the test statistic it is clear that at 98% t test we reject the null hypothesis on b2 and accept the null hypothesis on b3, for this reason therefore b3 is not statistically significant, therefore it is clear that ulc effects the p levels but for the umc it is not statistically significant and therefore it is not statistically correct to say that it affects the level of p.
Below is the summary of the test
t critical (98%) null hypothesis
Hypothesis test that b2+b3 = 1
We test the hypothesis by first stating the null and alternative hypothesis:
H0: b2+b3 = 0 or b2=b3=0
Ha: b2+b3 – 1 ≠ 0
DEGREES OF FREEDOM
T CALCULATED F CRITICAL (98%)
from the test the value of ess/degrees of freedom is greater than the value of rss/ degrees of freedom, the larger the value of f then the more significant is our values, this is because the variations in the model are more explained by ess, as the ess increase then the f value becomes increasingly large and leads us to reject the null hypothesis. Therefore we can conclude that the b1 is not equal to b2 which is not equal to zero.
From the above statistical test it is clear that the value of f derived is very large, this will lead to the rejection of the null hypothesis which states that b2 and b3 are equal to zero, for this reason therefore we can conclude that the value of b2 and b3 are not equal to zero and that they are statistically significant.
The larger the value of f the more statistical significant our parameters under question, the higher the f value means that the variation in the dependent variable are largely explained by the independent variable and the stochastic variable is not large in our model. For this reason we reject the null hypothesis and conclude that b2 plus b3 is not equal to zero but equal to one.
P. Schmidt (1976) Econometrics: an introduction to econometrics, Marcel Decker publishers, Ne w York
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