###### Downside Deviation

When trying to estimate downside risk (i.e., returns below the mean), we can... **Read More**

The classic normal linear regression model assumptions are as follows:

**I.** The relationship between the dependent variable, *Y*, and the independent variable, *X*, is **linear**.

A linear relationship implies that the change in *Y* due to a one unit change in *X* is constant, regardless of the value taken by *X*. If the relationship between the two is not linear, the regression model will not capture the trend accurattely, a situation that will result in inaccurate predictions. The model will be biased and either underestimate or overestimate *Y* at various points. For example, the model \(Y = \beta_{0}+\beta_{1}e^{\beta_{1}x}\) is non linear in \(β_{1}\), and therefore we should not attempt to fit a linear model between *X* and* Y.*

It also follows that the independent variable, *X*, must be non stochastic (must not be random). A random independent variable rules out a linear relationship between the dependent and independent variables.

In addition, linearity means the residuals should not exhibit a discernible pattern when plotted against the independent variable but should instead be completely random. In the example below, we’re looking at a scenario where the residuals appear to show a pattern when plotted against the independent variable, *X*. This effectively serves as evidence of a non-linear relation.

**II.** The expectation of the error terms is **zero**.

$$E(\epsilon)=0$$

**III. **The error terms (residuals) must be normally distributed.

A histogram of the residuals can be used to detect if the error term is normally distributed. A symmetric bell-shaped histogram indicates that the normality assumption is likely to be true.

**IV. **The variance of the error terms is **constant** across all observations.

$$E(\epsilon_i^2)=\sigma_{\epsilon}^2,\ i=1,2, …, n$$

This assumption is also known as the **homoskedasticity** assumption.

In case residuals and the predicted values increase simultaneously, then such a situation is known as **heteroscedasticity **(or heteroskedasticity).

To test for heteroscedasticity, one ought to plot the least square residuals against the independent variable. If there is an evident pattern in the plot, then that is a manifestation of heteroskedasticity.

**V.** The error terms, \(\epsilon\), must be **uncorrelated** across all observations.

$$E(\epsilon_i\epsilon_j)=0,\ \forall i\neq j$$

To verify this assumption, one should use a residual time series plot, which is a plot of residuals versus time. Fluctuating patterns around zero will indicate that the error term is dependent.