The process of determining a regression or prediction equation to predict Y from X , with all the method of least squares. In the resulting regression line, the sum of the squared discrepancies between the actual dependent values and the corresponding values predicted by the line are as small as possible, hence the name ‘least squares'” (Hassard, 1991). The estimated regression equation is: Y = ß0 + ß1X1 + ß2X2 + ß3D + ê Where the ßs are the OLS estimates of the Bs. OLS minimizes the sum of the squared residuals
The residual, ê, is the difference between the actual Y and the predicted Y and has a zero mean. In other words, OLS calculates the slope coefficients so that the difference between the predicted Y and the actual Y is minimized. The residuals are squared so as to compare negative errors to positive errors more easily. The properties are: 1. The regression line defined by 1 and 2 passes through the means of the observed values 2. The mean of the predicted Y’s for the sample will equal the mean of the observed Ys for the sample. 3. The sample mean of the residuals will be 0. 4. The correlation between the residuals and the predicted values of Y will be 0. 5. The correlation between the residuals and the observed values of X will be 0.
Stationarity can be defined as a time series yt is covariance (or weakly) stationary if, in support of if, its mean and variance are both finite and outside of time, and the auto-covariance doesn’t overgrow time, for those t and t-s, 1. Finite mean E (yt) = E (yt-s) = Âµ 2. Finite variance Var (yt) = E [(yt-Âµ) 2] = E [(yt-s – Âµ) 2] = 3. Finite auto-covariance Cov (yt, yt-s) = E [(yt-Âµ) (yt-s – Âµ)] = ÃŽÂ³s
The variance is time dependent and visits infinity as time strategies to infinity. A time series which is not stationary depending on mean can be done stationary by differencing. Differencing is a popular and effective method of removing a stochastic trend from a series. Nonstationarity in a time series occurs individuals no constant mean, no constant variance or those two properties. It could possibly originate from various sources nevertheless the most crucial one is the unit root.
Any sequence that contains one or more characteristic roots which can be comparable to is known as a unit root process. The most convenient model which will contain a unit root may be the AR (1) model. Look at the autoregressive process of order one, AR (1), below Yt = Ã‰Â¸Yt-1 + ÃŽÂµt Where ÃŽÂµt denotes a serially uncorrected white-noise error term which has a mean of zero and also a constant variance If Ã‰Â¸ = 1, becomes a random walk without drift model, that is certainly, a nonstationary process.
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