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Statistical Analysis

Statistical analysis is a component of data analytics.

In the context of business intelligence (BI), statistical analysis involves collecting and scrutinizing every data sample in a set of items from which samples can be drawn. A sample, in statistics, is a representative selection drawn from a total population.

Statistical analysis can be broken down into five discrete steps, as follows:

1-Describe the nature of the data to be analyzed.
2-Explore the relation of the data to the underlying population.
3-Create a model to summarize understanding of how the data relates to the underlying population.
4-Prove (or disprove) the validity of the model.
5-Employ predictive analytics to run scenarios that will help guide future actions.

TRY NOT TO CONFUSE THEM BOTH,DATA IS DIFFERENT FROM INFORMATION.

Reference to full study of OLS see below
https://statisticsbyjim.com/regression/ols-linear-regression-assumptions/

Some of the assumptions of ORDINARY LEAST SQUARE (OLS) .

OLS Assumption 1: The regression model is
linear in the coefficients and the error term

This assumption addresses the functional form of the model. In statistics, a regression model is linear when all terms in the model are either the constant or a parameter multiplied by an independent variable. You build the model equation only by adding the terms together. These rules constrain the model to one type:

In the equation, the betas (βs) are the parameters that OLS estimates. Epsilon (ε) is the random error.

In fact, the defining characteristic of linear regression is this functional form of the parameters rather than the ability to model curvature. Linear models can model curvature by including nonlinear variables such as polynomials and transforming exponential functions.

To satisfy this assumption, the correctly specified model must fit the linear pattern.

OLS Assumption 2: The error term has a population mean of zero

The error term accounts for the variation in the dependent variable that the independent variables do not explain. Random chance should determine the values of the error term. For your model to be unbiased, the average value of the error term must equal zero.

Suppose the average error is +7. This non-zero average error indicates that our model systematically underpredicts the observed values. Statisticians refer to systematic error like this as bias, and it signifies that our model is inadequate because it is not correct on average.

Stated another way, we want the expected value of the error to equal zero. If the expected value is +7 rather than zero, part of the error term is predictable, and we should add that information to the regression model itself. We want only random error left for the error term.

You don’t need to worry about this assumption when you include the constant in your regression model because it forces the mean of the residuals to equal zero. For more information about this assumption, read my post about the regression constant.

OLS Assumption 3: All independent variables are uncorrelated with the error term

If an independent variable is correlated with the error term, we can use the independent variable to predict the error term, which violates the notion that the error term represents unpredictable random error. We need to find a way to incorporate that information into the regression model itself.

This assumption is also referred to as exogeneity. When this type of correlation exists, there is endogeneity. Violations of this assumption can occur because there is simultaneity between the independent and dependent variables, omitted variable bias, or measurement error in the independent variables.

Violating this assumption biases the coefficient estimate. To understand why this bias occurs, keep in mind that the error term always explains some of the variability in the dependent variable. However, when an independent variable correlates with the error term, OLS incorrectly attributes some of the variance that the error term actually explains to the independent variable instead. For more information about violating this assumption, read my post about confounding variables and omitted variable bias.

OLS Assumption 4: Observations of the error term are uncorrelated with each other

One observation of the error term should not predict the next observation. For instance, if the error for one observation is positive and that systematically increases the probability that the following error is positive, that is a positive correlation. If the subsequent error is more likely to have the opposite sign, that is a negative correlation. This problem is known both as serial correlation and autocorrelation.

8.1 INTRODUCTION: AUTOCORRELATION

One of the standard assumptions in the regression model is that the error terms Eiand E j, associated with the ith and jth observations, are uncorrelated. Correlation in the error terms suggests that there is additional information in the data that has not been exploited in the current model. When the observations have a natural
sequential order, the correlation is referred to as autocorrelation.
Autocorrelation may occur for several reasons. Adjacent residuals tend to be
similar in both temporal and spatial dimensions. Successive residuals in economic
time series tend to be positively correlated. Large positive errors are followed
by other positive errors, and large negative errors are followed by other negative
errors. Observations sampled from adjacent experimental plots or areas tend to have
residuals that are correlated since they are affected by similar external conditions.
The symptoms of autocorrelation may also appear as the result of a variable
having been omitted from the right-hand side of the regression equation. If suc￾cessive values of the omitted variable are correlated, the errors from the estimated
model will appear to be correlated. When the variable is added to the equation, the
apparent problem of autocorrelation disappears. The presence of autocorrelation
has several effects on the analysis. These are summarized as follows:

THE PROBLEM OF CORRELATED ERRORS

1. Least squares estimates of the regression coefficients are unbiased but are not
efficient in the sense that they no longer have minimum variance.
2. The estimate of (72 and the standard errors of the regression coefficients may
be seriously understated; that is, from the data the estimated standard errors
would be much smaller than they actually are, giving a spurious impression
of accuracy.
3. The confidence intervals and the various tests of significance commonly
employed would no longer be strictly valid.
The presence of autocorrelation can be a problem of serious concern for the pre￾ceding reasons and should not be ignored.
We distinguish between two types of autocorrelation and describe methods for
dealing with each. The first type is only autocorrelation in appearance. It is due
to the omission of a variable that should be in the model. Once this variable is
uncovered, the autocorrelation problem is resolved. The second type of autocor￾relation may be referred to as pure autocorrelation. The methods of correcting for
pure autocorrelation involve a transformation of the data. Formal derivations of the
methods can be found in Johnston (1984) .