Estimator properties

### Least Squares Method – The method by which Excel calculates the values ​​of ( beta ) and ( alpha ) is called the least squares method. The calculation is done using formulas developed by mathematicians.
– The formulas themselves are called **estimates**, and the values ​​of ( beta ) and ( alpha ), calculated according to these formulas from the specific sample, are called **estimates**.
– The estimates according to the least squares method are called ordinary least squares (OLS) estimates. ### Properties of the estimates – For each sample, we will use the same formula to calculate (beta) and (alpha). That is, the estimates do not change from sample to sample (the formulas do not depend on the sample). – The estimates, on the other hand, may change from sample to sample. That is, the estimates yield values ​​that are random variables because the estimates calculated from them are sample-dependent. Each sample will give a different value. ### Statistical properties of the least squares estimates 1. They are unbiased.
2. Their variance is small compared to estimates by other methods. #### Explanation of the properties – **Least squares estimates are unbiased**: If we use least squares estimates to calculate ( beta ), then the probability that ( hat{beta} ) that we get will be **larger** than the true ( beta ), the same as the chance that it will be **smaller** than it. In other words: the center of the bell distribution of ( hat{beta} ) is in the true ( beta ). – **Least squares estimators have relatively small variance**: The variance of least squares estimators is relatively small. This causes the estimates obtained in different samples to be mostly scattered in a small area on the number line. #### Demonstrating the properties in charts – If an estimator is both unbiased and has a small variance, then the estimate obtained from it is likely to be relatively accurate. – For example, suppose that ( beta = 3 ), and the estimator we use to calculate ( beta ) is an unbiased estimator with low variance. The chart illustrates this:
– According to the chart, because the variance is small, in 95% of cases (hat{beta} ) that will be obtained will be between 2.9 and 3.1. And if the true value is 3, then the estimated value is not far from it and reflects it well. – Let us now look at the case where the estimator is biased (not unbiased), for example, let us assume that the center of the distribution is at 3.5 (and not at 3), and the variance is as in the previous diagram.
– According to the chart, in 95% of cases (hat{beta} ) that will be obtained will be between 3.4 and 3.6. Since the true value is 3, the estimated value is far from the true value and does not reflect it well. – Let’s look at the case where the estimator is unbiased but has a large variance.
– According to the chart, in 95% of cases (hat{beta} ) that will be obtained will be between 0.5 and 5.5. Since the true value is 3, the estimated value is far from the true value and does not reflect it well. ### In summary, the least squares estimates are unbiased with small variance (as in the first figure), and therefore the estimates derived from them (( hat{beta} ) and ( hat{alpha} )) relatively accurately reflect the true values ​​(( beta ) and ( alpha )).