Assignment Code: BECC-110 / ASST/BECC 110/ 2021-22

Course Code: BECC-110

Assignment Name: INTERMEDIATE MACROECONOMICS - II

Year: 2021 - 2022

Verification Status: Verified by Professor

Assignment I

Answer the following Descriptive Category Questions in about 500 words each. Each question

carries 20 marks. Word limit does not apply in the case of numerical questions. 2 × 20 = 40

Q1) What is meant by heteroscedasticity? Describe any one method of testing for the presence of heteroscedasticity.

Ans) The statistical term heteroscedastic (from the Ancient Greek words hetero "different" and skedasis, "dispersion") is used to describe a vector of random variables in which the variability of the random disturbance varies across distinct parts of the vector. The variance, or any other measure of statistical dispersion, might be used to quantify variability in this situation. As a result, heteroscedasticity can be defined as the absence of homoscedasticity. A classic example is a collection of observations about income in various cities around the world.

The presence of heteroscedasticity in regression analysis and the analysis of variance is a major source of worry because it invalidates statistical tests of significance that assume that the modelling mistakes all have the same variance. Even if the ordinary least squares estimator is still unbiased in the presence of heteroscedasticity, it is wasteful, therefore generalised least squares should be employed in place of ordinary least squares.

Informal test to detect the presence of heteroscedasticity

SPEARMAN’S RANK CORRELATION TEST

The presence of heteroscedasticity is detected by the researcher using a formal test called Spearman's rank correlation test, which is performed by a statistician. The following is an example of how to apply this test. Consider the following scenario: the researcher uses a straightforward linear model, Yi = ß0 + ß1Xi + ui, to discover heteroscedasticity. In order to discover heteroscedasticity in the data, the researcher must first obtain the absolute value of the residual and then rank the residuals in an ascending or descending order. Following that, the researcher computes the Spearman's rank correlation coefficient to determine heteroscedasticity in the data. Moving on to the procedure of detecting heteroscedasticity, the population rank correlation coefficient is considered to be zero, and the sample size is expected to be bigger than eight. In order to detect heteroscedasticity, a statistical significance test is performed. if the computed value of t is greater than the tabulated value, then the researcher can conclude that heteroscedasticity exists in the data set. The data, on the other hand, does not show any signs of heteroscedasticity.

It has following steps:

1. Fit the regression of Y on X and obtain the residuals.

2. Compute the Spearman’s rank correlation between absolute value of residuals and Xi (or Ŷi)

3. Test the null hypothesis that population correlation coefficient is zero using t-test. If the hypothesis is rejected, then heteroscedasticity is said to be present.

4. The t-statistics is given by:  rs = Spearman’s rank correlation coefficient.

Q2) What is meant by autocorrelation? Explain how Durbin-Watson test can be used to test for the presence of autocorrelation.

Ans) In mathematics, autocorrelation represents the degree of similarity between a particular time series and a lagged version of itself over a series of succeeding time periods. Even while autocorrelation is theoretically similar to correlation between two independent time series, autocorrelation employs the same time series twice: once in its original form, and once with one or more time periods lagging.

Consider the following scenario: If it's raining today, the data implies that it's more likely that it will rain tomorrow than if it's clear. For example, when it comes to investment, a stock can have a strong positive autocorrelation of returns, which means that if it's "up" today, it's more likely to be "up" tomorrow, as well, if it's "up" today.

Autocorrelation can, without a doubt, be a beneficial tool for traders, and it is particularly useful for technical analysts in particular.

Main Points

1. Autocorrelation represents the degree of similarity between a given time series and a lagged version of itself over successive time intervals.

2. Autocorrelation measures the relationship between a variable's current value and its past values.

3. An autocorrelation of +1 represents a perfect positive correlation, while an autocorrelation of negative 1 represents a perfect negative correlation.

4. Technical analysts can use autocorrelation to measure how much influence past prices for a security have on its future price.

It is a measure of autocorrelation (also known as serial correlation) in residuals from regression analysis that is developed by Durbin and Watson. The resemblance of a time series over repeated time intervals is referred to as autocorrelation. It can result in underestimations of the standard error of the mean and can lead you to believe that predictors are important when they are not. The Durbin Watson test searches for a specific sort of serial correlation, known as the AR(1) process, when doing the analysis.

The Hypotheses for the Durbin Watson test are:

H0 = no first order autocorrelation.

H1 = first order correlation exists.

(For a first order correlation, the lag is one time unit).

Assumptions are:

That the errors are normally distributed with a mean of 0.

The errors are stationary.

The test statistic is calculated with the following formula:

Where Et are residuals from an ordinary least square’s regression.

The Durbin Watson test reports a test statistic, with a value from 0 to 4, where:

2 is no autocorrelation.

0 to <2 is positive autocorrelation (common in time series data).

>2 to 4 is negative autocorrelation (less common in time series data).

A rule of thumb is that test statistic values in the range of 1.5 to 2.5 are relatively normal. Values outside of this range could be cause for concern. Field(2009) suggests that values under 1 or more than 3 are a definite cause for concern.

Assignment II

Answer the following Middle Category Questions in about 250 words each. Each question

carries 10 marks. Word limit does not apply in the case of numerical questions. 3 × 10 = 30

Q3) Explain why an error variable is added to the regression model. Differentiate between

the error term (u) and the residual (𝑢).

Ans) In statistics, errors-in-variables models or measurement error models are regression models that account for measurement errors in the independent variables. In contrast, standard regression models assume that those regressors have been measured exactly, or observed without error; as such, those models account only for errors in the dependent variables, or responses.

Regression is a maximum likelihood estimation where we find parameters of the relation between independent and dependent variables (which is in the form of an equation often times) which maximize the likelihood of getting such samples from the population.

Since regression is an estimation, we cannot be completely correct at it. So, the error term is a catch-all for what we miss out in this estimation because in reality

1. The true relation may not be what we assume(linear relation in case of linear regression)

2. There may be other variables not included in the model that cause variation in response variable

3. There may be measurement errors in the observations.

Difference between error terms and residuals

The discrepancy between an observation and its predicted value is commonly defined as error in statistics. An observation's estimated value is called a residual.

As a result, statistical mistakes are primarily theoretical, but residuals can be estimated using applicable statistics. Parameters are numbers that apply to a population. For example, population height has a true mean value, which represents the population's mean height. Instead of surveying the entire population, we employ a statistical measure, the mean height of a random sample.

The error is the difference between an individual's height and the population's mean. The residual is the height difference.

A true regression model yields an output value based on an input value, while (s). The error is the difference between an actual and predicted result from a valid regression. The sample regression is calculated using sample data. As a result, this sample regression's residual refers to the difference between an observed and estimated output value.

Q4) When do you encounter the problem of multicollinearity? What are the remedial measures for the problem of multicollinearity?

Ans) Multicollinearity occurs when independent variables in a regression model are correlated. This correlation is a problem because independent variables should be independent. If the degree of correlation between variables is high enough, it can cause problems when you fit the model and interpret the results.

Remedial Measures of  multicollinearity:

Multicollinearity does not actually bias results; it just produces large standard errors in the related independent variables. With enough data, these errors will be reduced.

In a pure statistical sense multicollinearity does not bias the results, but if there are any other problems which could introduce bias multicollinearity can multiply ( by orders of magnitude ) the effects of that bias. More importantly, the usual use of regression is to take coefficients from the model and then apply them to other data. If the new data differs in any way from the data that was fitted, we may introduce large errors in predictions because the pattern of multicollinearity between the independent variables is different in new data from the data used for your estimates. We try seeing what happens if we use independent subsets of your data for estimation and apply those estimates to the whole data set.

In addition, we may:

1. Persist in multicollinearity. Multicollinearity does not affect the fitted model if the predictor variables are multicollinear like the data.

2. Drop a variable. To get a model with significant coefficients, remove an explanatory variable. But you lose data (by dropping a variable). Missing a relevant variable leads to biased coefficient estimations for the rest.

3. Get more info. This is the best option. Data can help refine parameter estimates (with lower standard errors).

4. Norm-center the predictors. This has no impact on the findings of a regression. However, if a correctly developed computer programme is not employed, it can help overcome rounding and other computational issues.

5. Standardize your variables. Reduces false-positive condition indexes above 30.

Q5) Interpret the coefficient of determination (𝑅2). Distinguish between 𝑅2 and adjusted- 𝑅2.

Ans) The coefficient of determination is the square of the correlation (r) between predicted y scores and actual y scores; thus, it ranges from 0 to 1.

With linear regression, the coefficient of determination is also equal to the square of the correlation between x and y scores.

An R2 of 0 means that the dependent variable cannot be predicted from the independent variable.

An R2 of 1 means the dependent variable can be predicted without error from the independent variable.

An R2 between 0 and 1 indicates the extent to which the dependent variable is predictable. An R2 of 0.10 means that 10 percent of the variance in Y is predictable from X; an R2 of 0.20 means that 20 percent is predictable; and so on.

The formula for computing the coefficient of determination for a linear regression model with one independent variable is given below.

Coefficient of determination. The coefficient of determination (R2) for a linear regression model with one independent variable is:

R2 = { ( 1 / N ) * Σ [ (xi - x) * (yi - y) ] / (σx * σy ) }2

where N is the number of observations used to fit the model, Σ is the summation symbol, xi is the x value for observation i, x is the mean x value, yi is the y value for observation i, y is the mean y value, σx is the standard deviation of x, and σy is the standard deviation of y.

R-Squared vs. Adjusted R-Squared

R-squared and adjusted R-squared enable investors to measure the performance of a mutual fund against that of a benchmark. Investors may also use them to calculate the performance of their portfolio against a given benchmark.

Adjusted R-squared can provide a more precise view of that correlation by also considering how many independent variables are added to a particular model against which the stock index is measured. This is done because such additions of independent variables usually increase the reliability of that model—meaning, for investors, the correlation with the index.

Assignment III

Answer the following Short Category Questions in about 100 words each. Each question carries 6 marks. 5 ×6 = 30

Q6) What is a dummy variable? Illustrate the situation of dummy variable trap.

Ans) A dummy variable[a] is a variable that only accepts the values 0 or 1 to indicate the existence or absence of a categorical influence that may affect the outcome. In a regression model, they filter data into mutually exclusive categories.

A dummy independent variable (sometimes called a dummy explanatory variable) with a value of 0 has no effect on the dependent variable, while a dummy with a value of 1 affects the intercept.

Assume, for example, that group membership is a qualitative variable in a regression. If group membership is assigned a value of 1, all others are assigned a value of 0. For non-members, the intercept is the constant term plus the coefficient of the membership dummy.

Q7) Describe the properties of OLS estimators.

Ans) The OLS estimators are linear (not curved), unbiased (average out the same as the data they represent) and have less variation than other models. BLUE stands for Best (least variance) Linear Unbiased Estimator.

These qualities make OLS estimators attractive because they operate well. However, some assumptions must be met, such as random data sampling and non-total collinearity.

Because OLS estimators are linear, the resulting model should be a line. Unlike the assumption that the underlying data is linear. The unbiased property suggests that the OLS estimators do not consistently skew the data. Because OLS has the least variance, it is more efficient than alternatives, even if they are unbiased. It is more precise than other methods.

Q8) Describe the procedure of applying RESET test.

Ans) The RESET test is a popular diagnostic for correctness of functional form. The basic assumption is that under the alternative the model can be written in the form y=X * beta + Z * gamma. Z is generated by taking powers either of the fitted response, the regressor variables, or the first principal component of X. A standard F-Test is then applied to determine whether these additional variables have significant influence. The test statistic under H_0 follows an F distribution with parameter degrees of freedom.

This function was called reset in previous versions of the package. This interface is currently still included, but a warning is issued. Please use reset test instead.

Q9) In the context of hypothesis testing distinguish between one-tailed and two-tailed tests. Use appropriate diagram.

Ans) Key Differences Between One-tailed and Two-tailed Test

The fundamental differences between one-tailed and two-tailed test, is explained below in points:

One-tailed test, as the name suggest is the statistical hypothesis test, in which the alternative hypothesis has a single end. On the other hand, two-tailed test implies the hypothesis test; wherein the alternative hypothesis has dual ends.

In the one-tailed test, the alternative hypothesis is represented directionally. Conversely, the two-tailed test is a non-directional hypothesis test.

In a one-tailed test, the region of rejection is either on the left or right of the sampling distribution. On the contrary, the region of rejection is on both the sides of the sampling distribution.

A one-tailed test is used to ascertain if there is any relationship between variables in a single direction, i.e., left or right. As against this, the two-tailed test is used to identify whether or not there is any relationship between variables in either direction.

In a one-tailed test, the test parameter calculated is more or less than the critical value. Unlike, two-tailed test, the result obtained is within or outside critical value.

When an alternative hypothesis has ‘≠’ sign, then a two-tailed test is performed. In contrast, when an alternative hypothesis has ‘> or <‘ sign, then one-tailed test is carried out.

Q10) Write a short note on the types of specification errors in a regression model.

Ans) There are two basic types of specification errors. In the first, we mis specify a model by including in the regression equation an independent variable that is theoretically irrelevant. In the second, we mis specify the model by excluding from the regression equation an independent variable that is theoretically relevant.

Types of Specification Errors:

The four common sources of errors in specification: functional form, autocorrelated disturbances, heteroscedasticity, and missing variables. The Savin-White test, as well as the standard tests for only one source of error, are special cases of the test developed. The generalized test is applied to data in an illustrative example.