Autocorrelation

Autocorrelation occurs due to the chance correlation of the error term of a particular household with some other household or firm. Autocorrelation is also named chance correlation. Autocorrelation is also applied in the case of time series analysis.

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The process of autocorrelation is defined as the correlation that exists between the members of the series of the observations that are planned with respect to time.

If two types of data are considered – a cross sectional type of data and a time series type of data—then for the cross sectional type of data, if the change in the income of a particular person affects the consumption expenditure of another household other than his, then autocorrelation is present in the data. Similarly, for the time series type of data, if an output is low in one quarter due to a labor strike, and if the data showing low output continues in the next quarter as well, then autocorrelation is supposed to be present in the data.

The process of autocorrelation is defined as the type of lag correlation for a given type of series with itself, which is lagged by several numbers of time units. On the other hand, serial autocorrelation is that type of autocorrelation that is defined as the process of lag correlation between two series in time series data.

There are certain patterns that are exhibited by autocorrelation.

Autocorrelation exhibits patterns among the residual errors. Autocorrelation also occurs in cases when the error shows a cyclical kind of pattern, etc.

The major reason why autocorrelation occurs is because of the inertia or sluggishness that is present in time series data.

The occurrence of the non stationary property in time series data also gives rise to the phenomenon of autocorrelation. Thus, to make the time series almost free of the problem of autocorrelation, the researcher should always make the data stationary.

The researcher should know that autocorrelation can be positive as well as negative. Economic time series generally exhibits positive autocorrelation as the series moves in an upward or downward pattern. If the series moves in a constant upward and downward movement, then autocorrelation is negative.

The major consequence of using ordinary least square (OLS) in the presence of autocorrelation is that it will simply make the estimator inefficient. As a result, the hypothesis testing procedures will give inaccurate results due to the presence of autocorrelation.

There is a popular test called the Durbin Watson test that detects the presence of autocorrelation. This test is conducted under the null hypothesis that there is no autocorrelation in the data. A test statistic called ‘d’ is computed, which is defined as the ratio between the sum of the square of the difference in the residuals with ith and (i-1) time and the square of the residual in ith time. If the upper critical value of the test comes out to be less than the value of ‘d,’ then there is no autocorrelation. If the lower critical value of the test is more than the value of ‘d,’ then there is autocorrelation.

If one detects autocorrelation in the data, then the first thing a researcher should do is that he should try to find whether or not the autocorrelation is pure. If it is pure autocorrelation, then one can transform it into the original model, which is free from pure autocorrelation.

Canonical Correlation

A canonical correlation is a correlation between two canonical or latent types of variables. In canonical correlation, one variable is an independent variable and the other variable is a dependent variable. It is important for the researcher to know that unlike regression analysis, in canonical correlation, the researcher can find a relationship between many dependent and independent variables. A statistic called the Wilk’s Lamda is used for testing the significance of the canonical correlation. The work of the canonical correlation is the same as in simple correlation. In both of these, the point is to provide the percentage of the variances in the dependent variable that are explained by the independent variable. So, canonical correlation is defined as the tool that measures the degree of the relationship between the two variates.

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The process of canonical correlation is considered the member of the multiple general linear hypotheses, and therefore the assumptions of multiple regressions are also assumed in canonical correlation as well.

There are concepts and terms associated with canonical correlation. These concepts and terms will help a researcher better understand canonical correlation. They are as follows:

1. Canonical variable or variate: A canonical variable in canonical correlation is defined as the linear combination of the set of original variables. These variables in canonical correlation are a form of latent variables.

2. Eigen values: The value of the Eigen values in canonical correlation are considered as approximately being equal to the square of the value of the canonical correlation. The Eigen values basically reflect the proportion of the variance in the canonical variate, which is explained by the canonical correlation that relates to the two sets of variables.

3. Canonical Weight: The other name for canonical weight is the canonical coefficient. The canonical weight in canonical correlation must first be standardized. It is then used to assess the relative importance of the contribution of the individual’s variable.

4. Canonical communality coefficient: This coefficient in canonical correlation is defined as the sum of the squared structure coefficients for the given type of variable.

5. Redundancy coefficient, d: This coefficient in canonical correlation basically measures the percent of the variance of the original variables of one set that is predicted from the other set through canonical variables.

6. Likelihood ratio test: This significance test in canonical correlation is used to carry out the significance test of all the sources of the linear relationship between the two canonical variables.

There are certain assumptions that are made by the researcher for conducting canonical correlation. They are as follows:

1. It is assumed that the interval type of data is used to carry out canonical correlation.

2. It is assumed in canonical correlation that the relationships should be linear in nature.

3. It is assumed that there should be low multicollinearity in the data while performing canonical correlation. If the two sets of data are highly inter-correlated, then the coefficient of the canonical correlation is unstable.

4. There should be unrestricted variance in canonical correlation. If the variance is not unrestricted, then this might make the canonical correlation look unstable.

Most researchers think that canonical correlation is computed in SPSS. However, canonical correlation is obtained while computing MANOVA in SPSS. In MANOVA, canonical correlation is used in data sets where one refers to the one set of variables as the dependent and the other as the covariates.

Chi Square test

Parametric tests are those kinds of tests that involve the use of parameters, and the chi square test is a parametric tests.

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There are varieties of chi square tests that are used by the researcher. They are cross tabulation, chi square test for the goodness of fit, likelihood ratio test, chi square test, etc.

The task of the chi square test is to test the statistical significance of the observed relationship with respect to the expected relationship. The chi square statistic is used by the researcher for determining whether or not a relationship exists.

In the chi square test, the null hypothesis is assumed as there not being an association between the two variables that are observed in the study. The chi square test is calculated by evaluating the cell frequencies that involve the expected frequencies in those types of cases when there is no association between the variables. The comparison between the expected type of frequency and the actual observed frequency is then made in the chi square test. The computation of the expected frequency in the chi square test is calculated as the product of the total number of observations in the row and the column, which is divided by the total size of the sample.

The calculation of the chi square type of statistic in the chi square test is done by computing the sum of the square of the deviation between the observed and the expected frequency, which is divided by the expected frequency.

The researcher should know that the greater the difference between the observed and expected cell frequency, the larger the value of the chi square statistic in the chi square test.

In order to determine if the association between the two variables exists, the probability of obtaining a value of chi square should be larger than the one obtained from the chi square test of cross tabulation.

There is one more popular test called the chi square test for goodness of fit.

This type of chi square test called the chi square test for goodness of fit helps the researcher to understand whether or not the sample drawn from a certain population has a specific distribution and whether or not it actually belongs to that specified distribution. This type of chi square test can be applicable to only discrete types of distribution, like Poisson, binomial, etc. This type of chi square test is an alternative test for the non parametric test called the Kolmogorov Smrinov goodness of fit test.

The null hypothesis assumed by the researcher in this type of chi square test is that the data drawn from the population follows the specified distribution. The chi square statistic in this chi square test is defined in a similar manner to the definition in the above type of test. One of the important points to be noted by the researcher is that the expected number of frequencies in this type of chi square test should be at least five. This means that the chi square test will not be valid for those whose expected cell frequency is less than five.

There are certain assumptions in the chi square test.

The random sampling of data is assumed in the chi square test.

In the chi square test, a sample with a sufficiently large size is assumed. If the chi square test is conducted on a sample with a smaller size, then the chi square test will yield inaccurate inferences. The researcher, by using the chi square test on small samples, might end up committing a Type II error.

In the chi square test, the observations are always assumed to be independent of each other.

In the chi square test, the observations must have the same fundamental distribution.