(4) Discuss necessary assumptions regarding the error terms, specifically about
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(4) Discuss necessary assumptions regarding the error terms, specifically about the probability distribution of this term and the rationale for this assumption Estimation and hypothesis testing are the two main branches of statistical inference. Previously, we used the method of OLS to estimate the parameters of PRM. Notice that is a one number (point) estimate of the unknown population valueP.. Because of sampling fluctuations, a single estimate is likely to differ from the true value, although in repeated sampling its mean value is expected to be equal to the true value, E(,)-A, in another word, how good the computed SRF is as an estimate of the true PRF. Now, we need hypothesis testing (formal testing procedure) to reject or not reject the null hypothesis. Without the knowledge of the sampling distributions of OLS estimators, we will not be able to engage in hypothesis testing. To derive the sampling distribution of the OLS estimators P, andP., we need to discuss one assumption about the nature of the error term" from the list of assumptions of the CLRM. In the PRF Y,-, + ,X, +14, the error term follows the normal distribution with mean zero and variance o", That is.",-N(0,') The rationale for this assumption is supported by a celebrated theorem in statistics, the central limit theorem (CLT) Notice that "represents the sum of all the forces that affect Y but not specifically included in the regression model because there are so many of them and the individual effect of any one such force on Y may be too minor. If al these forces are random, by invoking the CLT, we can assume that "follows the normal distributionExplanation / Answer
1). The regression model is linear in parameters; it may or may not be linear in the variables. That is, the regression model is of the following type :-
Yi = B1 + B2Xi + Mui
2). The explanatory variable X is uncorrelated with the disturbance term Mu. However, if the X variable is stochastic(i.e its value is a fixed number), this assumption is automatically fulfilled. Even if the X value is stochastic, with a large enough sample size this assumption can be related without severely affecting the analysis.
3). Given the value of Xi, the expected or the mean value of the disturbance term Mu is zero . That is
E(Mu / Xi ) = 0
4). The variance of each Mui is constant, or homoscedastic (homo means equal and scedastic means variance). That is ; var(Mui) = sigma2
5). There is no correlation between between two terms. This assumption is the no autocorrelation.
This means; cov(Mui , Muii) = 0. Here cov stands for covariance , i and j are two types of error terms.
This equation means that there is no systematic relationship between the two error terms.
6). The regression model is correctly specified. Alternatively, there is no specification bias or specification error in the model used in empirical analysis.
This assumption implies thaat we have included all the variables that affect a particular phenomenon. Thus, if studying the demand for automobile, if we include only price of automobiles and consumer income and do not take into account variables like advertising, financing costs, and gasoline prices, we will be committing a model specification error.
7). The PRF Yi = B1 + B2X2 + Mui the error term Mui follows the normal distribution with mean zero and variance sigma2. That is; Mui = N(0, sigma2)
The rationale for this assumption is central limit theorem (CLT). CLT states that if there are a large number of independent and identically distributed random variables, then with a few exceptions, the distribution of their sum tends to be a normal distribution as the number of such variables increases infinitely.
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