1. Time Series Forecasting Models & Introduction to Eviews

2. Types of Data Primary Data Time Series Data Qualitative Data
Secondary Data
Time Series Data
Cross-sectional Data
Pooled Data
Panel Data
Qualitative Data
Quantitative Data
Qualitative Data: They represent some characteristics or attributes. They depict descriptions that may be observed but cannot be computed or calculated. For example, data on attributes such as intelligence, honesty, wisdom, cleanliness, and creativity collected using the students of your class a sample would be classified as qualitative. They are more exploratory than conclusive in nature.
Quantitative Data: These can be measured and not simply observed. They can be numerically represented and calculations can be performed on them. For example, data on the number of students playing different sports from your class gives an estimate of how many of the total students play which sport. This information is numerical and can be classified as quantitative.
Depending on the source, it can be classified as primary data or secondary data.
Time series data of a variable have a set of observations on values at different points of time. They are usually collected at fixed intervals, such as daily, weekly, monthly, annually, quarterly, etc. Time series econometrics has applications in macroeconomics, but mainly in financial economics where it is used for price analysis of stocks, derivatives, currencies, etc.
Cross-section data are collected at the same point of time for several individuals. Examples are opinion polls, income distribution, data on GNP per capita in all European countries, etc.
Pooled data is a mixture of time series data and cross-section data. One example is GNP per capita of all European countries over ten years.
Panel, longitudinal or micropanel data is a type that is pooled data of nature. The difference is that we measure over the same cross-sectional unit for individuals, households, firms, etc. This branch of econometrics is called microeconometrics.

3. Important Concepts Stochastic Process
Stochastic process is a collection of random variable ordered in time.
White Noise
A stochastic process { 𝑢 𝑡 } is called white noise and can be demostrated as:
{ 𝑢 𝑡 }~ WN (o, σ2) if
E( 𝑢 𝑡 ) = 0; i.e. expected value of 𝑢 𝑡 is equal to 0.
Var( 𝑢 𝑡 ) = σ2; i.e. constant/finite variance (the variance does not change over time).

4. Important Concepts Martingale Models
A martingale model is where the best expectation for t+1 is the value today at time t. i.e.
E(Yt/Ft) = Yt
Deterministic System
A deterministic system is a system in which no randomness is comprised in the development of future states of the system.
where; Ft is information set (contains all the information up to time t)

5. Important Concepts Stationary Stochastic Process
Characteristics of stationary stochastic process is that its distribution moments, mean, variance, covariance, should be time invariant i.e. should be independent of time.
A stochastic process having mean, variance and covariance constant over time;
E(Yt) = µ
E(Yt) = σ2
E[(Yt - µ) (Yt+k - µ) ] = θk -- Covariance Stationary

6. Important Concepts Non-stationary Stochastic Process / Random Walk
Lets,
Yt = αYt-1 + 𝑢 𝑡 ;
If α = 1, then it is a random walk process /non-stationary stochastic process
If α < 1, then it is a stationary process as the shocks vanishes in short run.
If α > 1, then it is an explosive / non-stationary process as shocks does not wipe out and have increasing effect on the series each period.

7. Important Concepts Random Walk Process Random Walk with Drift
Random Walk with Drift and Trend

8. Important Concepts Difference Stationary Process
Let’s say random walk process;
𝑦 𝑡 = 𝑦 𝑡−1 + 𝑢 𝑡 by subtracting y t−1 from both sides
y t − y t−1 = y t−1 + u t − y t−1
The difference stationary has been obtained as ∆ 𝑦 𝑡 = 𝑢 𝑡 𝐸 ∆y t =𝐸 𝑢 𝑡 = 𝑉𝑎𝑟 ∆y t =𝑉𝑎𝑟 𝑢 𝑡 = 𝜎 𝐶𝑜𝑣 ∆y t , ∆y t−1 =𝐶𝑜𝑣 𝑢 𝑡 , 𝑢 𝑡−1 =0

9. Hypothesis Testing
Hypothesis testing is an integral part of modelling.
An important characteristic of a good research question is that it can be converted into testable hypothesis.
There are mainly three types of tests that are used commonly:
Wald test (t-test & F-test)
LM test
LR test.
Testing hypothesis using more than one of the tests ensures reliable results

10. Hypothesis Testing Wald t-Test
Wald tests are the most commonly used tests for hypothesis testing.
Let’s assume a following model:
𝑦 𝑡 = 𝛽 𝛽 2 𝑥 𝑡 + 𝜖 𝑡 𝐻 0 : 𝛽 2 =0
𝐻 1 : 𝛽 2 ≠0
Test statistic:
𝑡−𝑠𝑡𝑎𝑡= 𝛽 2 − 𝛽 2 𝜎 𝛽 2 ~ 𝑡 (𝑛−𝑘)

11. Hypothesis Testing There are two ways to make the decision:
If t-stat > critical value ( 𝛼 2 , 𝑛−𝑘), reject the null hypothesis and conclude that beta coefficient is not equal to zero or vice versa.
If p-value < 𝛼, reject the null hypothesis and accept the alternate one or vice versa.
Applications in Eviews

12. Hypothesis Testing Wald F-Test
Wald F-test is used to test whether the presence or absence of independent variables makes any difference in a model.
The main model is referred to as ‘unrestricted model’. A ‘restricted model’ is made by removing the variables that needs to be tested. Let’s assume a following unrestricted model.
𝐺𝐷𝑃=𝑓(𝐶𝑂𝑁𝑆,𝐼𝑁𝑉,𝐺𝑂𝑉,𝐸𝑋,𝐼𝑀𝑃)
𝐺𝐷𝑃 𝑡 =𝛼+ 𝛽 1 𝐶𝑂𝑁𝑆 𝑡 + 𝛽 2 𝐼𝑁𝑉 𝑡 + 𝛽 3 𝐺𝑂𝑉 𝑡 + 𝛽 4 𝐸𝑋 𝑡 + 𝛽 5 𝐼𝑀𝑃 𝑡 + 𝜀 𝑡

13. Hypothesis Testing Wald F-Test
F-test uses the following test statistic to test the above hypothesis:
𝐹−𝑠𝑡𝑎𝑡. = 𝑅𝑆𝑆 𝑅 − 𝑅𝑆𝑆 𝑈𝑅 𝑔 𝑅𝑆𝑆 𝑈𝑅 𝑛−𝑘 ~ 𝐹 (𝑔,𝑛−𝑘)

14. Hypothesis Testing Wald F-Test
Just like t-test, there are two ways to make the decision:
If F-stat > critical value (𝑔, 𝑛−𝑘), reject the null hypothesis and conclude that exports and imports have significant effect on GDP.
If p-value < 𝛼, reject the null hypothesis and accept the alternate one or vice versa.
Applications in Eviews
Scalar fstat = formula
Scalar prob. = g, n-k)

15. Hypothesis Testing LR Test
LR test is also referred to as Likelihood Ratio test. It can also be used as an alternate to t- test and F-test.
Just like F-test, it put restrictions on the main model (also referred to as unrestricted model) to test various hypothesis. Assuming the same example as in the F-test, LR test uses the following test statistic to test the null hypothesis of 𝛽 4 = 𝛽 5 =0:
𝐿𝑅−𝑠𝑡𝑎𝑡. = −2 ( 𝐿𝑜𝑔𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑 𝑅𝑒𝑠𝑡. − 𝐿𝑜𝑔𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑 𝑢𝑛𝑟𝑒𝑠𝑡 ~ 𝜒 2 (𝑔)
Here, g is the number of restrictions.

16. Hypothesis Testing LR Test Applications in Eviews
Scalar lrstat=formula
Scalar
Where, g is the number of restrictions

17. Hypothesis Testing LM Test
The LM test, also referred to as Lagrange Multiplier test, can be used as alternate to the above discussed tests. There are two ways to perform this test:
Frist way: LM test statistic can be calculated using the following formula similar to F-test. However, LM test uses chi-square distribution instead of F-distribution.
𝐿𝑀−𝑠𝑡𝑎𝑡. = 𝑅𝑆𝑆 𝑅 − 𝑅𝑆𝑆 𝑈𝑅 𝑅𝑆𝑆 𝑅 𝑛 ~ 𝜒 2 (𝑔)

18. Hypothesis Testing LM Test
Second way: Take the residuals series from the restricted model and then regress it with all the right-hand side variables of the unrestricted model. This regression is referred to as ‘side regression’.
𝑒 𝑡 =𝛼+ 𝛽 1 𝐶𝑂𝑁𝑆 𝑡 + 𝛽 2 𝐼𝑁𝑉 𝑡 + 𝛽 3 𝐺𝑂𝑉 𝑡 + 𝛽 4 𝐸𝑋 𝑡 + 𝛽 5 𝐼𝑀𝑃 𝑡 + 𝜖 𝑡 𝐿𝑀−𝑠𝑡𝑎𝑡. =𝑛∗ 𝑅 2 ~ 𝜒 2 (𝑔) Here, n is the number of observations and 𝑅 2 is the coefficient of determination, that explains how much variation in the dependent variable is explained by the regressor or explanatory variables.

19. Hypothesis Testing LM Test Applications in Eviews
Scalar lmstat = formula
Scalar g)

20. ARIMA models One of the basic models in time series analysis
Past values of a variable are useful to predict the future values
ARMA models use past values of the dependent variables to model the observed time series
Helpful when the number and relations of the explanatory variables (independent variables) is not clear
Uses OLS (Ordinary Least Square) estimator to estimate the relation of a variable with its past values.

21. ARIMA models Model Equation
ARMA(p,q) model equation may consist of two parts: AR (Auto-regressive) part and MA (Moving Average) part. AR (p) model can be written as follows:
𝑦 𝑡 = 𝛼 0 + 𝛼 1 𝑦 𝑡−1 + 𝛼 2 𝑦 𝑡−2 +…+ 𝛼 2 𝑦 𝑡−2 + ∈ 𝑡 = 𝛼 0 + 𝑝=1 𝑛 𝛼 𝑝 𝑦 𝑡−𝑝 + ∈ 𝑡 ; ∈ 𝑡 ~ (0, 𝜎 2 )
MA (q) model can be written as follows:
𝑦 𝑡 = 𝛽 0 + 𝛽 1 𝜖 𝑡−1 + 𝛽 2 𝜖 𝑡−2 +…+ 𝛽 2 𝜖 𝑡−2 + 𝑒 𝑡 = 𝛽 0 + 𝑞=1 𝑛 𝛽 𝑞 𝜖 𝑡−𝑞 + 𝑒 𝑡 ; 𝑒 𝑡 ~ (0, 𝜎 2 )
ARMA (p,q) model can be written as follows:
𝑦 𝑡 = 𝛼 0 + 𝑝=1 𝑛 𝛼 𝑝 𝑦 𝑡−𝑝 + ∈ 𝑡 + 𝑞=1 𝑛 𝛽 𝑞 𝜖 𝑡−𝑞 + 𝑒 𝑡 ; 𝑒 𝑡 ~ (0, 𝜎 2 )

22. ARIMA models Hypothesis Testing
𝐻 0 : 𝛽 𝑛 =0; or 𝛼 𝑛 =0 ; no significant effect
𝐻 1 : 𝛽 𝑛 ≠0; or 𝛼 𝑛 ≠0 ; significant effect
Test Equation
𝑡−𝑠𝑡𝑎𝑡= 𝛽 𝑛 − 𝛽 𝑛 𝜎 𝛽 𝑛 ~ 𝑡 (𝑛−𝑘)

23. ARIMA models Decision Rule There are two ways to make the decision:
If t-stat > critical value ( 𝛼 2 , 𝑛−𝑘), reject the null hypothesis and conclude that beta coefficient is not equal to zero or vice versa.
If p-value < 𝛼, reject the null hypothesis and accept the alternate one or vice versa.

24. ARIMA models Assumptions
All the series used must be stationary at mean and variance
All the standard deterministic and stochastic assumptions of OLS regression
No Serial correlation
No heteroskedasticity

25. ARIMA models Box-Jenkins Approach for ARMA Modelling
Take ‘log’ of the original series and obtain AC (Autocorrelation) and PAC (Partial-autocorrelation) of the series.
If ACF is falling to zero gradually, it means the series is stationary and the AR model can used to model series. The number of spikes of PAC will tell the degree of AR model
If ACF has one or two spikes and remaining are ACs are zero or insignificant, then MA model might be suitable and the number of spikes of the ACF is the degree of MA model
If ACF is close to zero or insignificant, the series is random showing no relation between the observations of the series. ARMA model cannot be used.
If ACFs have high values after fix intervals, it means the series has a seasonal AR component.
If there is no decreasing trend in the ACFs, it means the series is non-stationary
If both ACF and PACFs falls to zero gradually, ARMA (p,q) model might be suitable to model the series.

26. ARIMA models
2. Fit an AR(p), MA(q), ARMA (p,q) or ARIMA(p,d,q) model to the time series. 3. Conduct residuals diagnostics to test if the residuals are uncorrelated and random. 4. Forecast using the fitted model.

27. ARIMA models
Applicatons in Eviews
Forecasting using ARMA models

28. Simple Linear Regression Analysis
The population regression function (PRF);
𝑌 𝑖 = 𝛽 0 + 𝛽 1 𝑋 𝑖 + 𝑢 𝑖
where 𝑌 is dependent variable; 𝑋 𝑖 is the explanatory variable; 𝑢 is the stochastic disturbance term; and finally 𝑖 is the 𝑖th observation.
The simple regression function (SRF);
𝑌 = 𝛽 𝛽 1 𝑋 𝑖 + 𝑢 𝑖
where 𝛽 0 is estimator of 𝛽 0 ; and 𝛽 1 is estimator of 𝛽 1 . If
𝑌 𝑖 = 𝑌 𝑖 + 𝑢 𝑖 , Then
𝑢 𝑖 = 𝑌 𝑖 − 𝑌 𝑖 and 𝑢 𝑖 = 𝑌 𝑖 − 𝛽 0 − 𝛽 1 𝑋 𝑖

29. Simple Linear Regression Analysis
Assumptions of OLS (Ordinary Least Squares Regression)
Deterministic Assumptions
𝑦 𝑡 is endogenous variable and 𝑥 𝑡 is exogeneous variable.
The time period ‘t’ comprises values from 1 to n, the total number of observations
The number of observations, n, must be greater than the number of parameters, k. i.e. n > k
Model is linear
Model is structured properly.

30. Simple Linear Regression Analysis
Assumptions of OLS (Ordinary Least Squares Regression)
Stochastic Assumptions
The residuals are normally distributed, i.e. 𝑒 𝑡 ∿ 𝑁(0, 𝛿 2 )
The residuals are homoscedastic. i.e. the variance of the residuals is constant.
There is no autocorrelation between the residuals.
The expected value of the residuals is zero, i.e. 𝐸 𝑒 𝑡 =0

31. Simple Linear Regression Analysis
Testing Deterministic Assumptions
In testing the deterministic assumptions, the following tests are used to check if the model is structured properly and all the necessary variables are included.
Coefficient of Determination – R2
The coefficient of determination, R2, shows the power of the independent variables to explain the variations in the dependent variable.
It is calculated 𝑅 2 is as follows,
𝑅 2 = 𝜌 𝑥𝑦 where, 𝜌 𝑥𝑦 is the correlation between the two variables.

32. Simple Linear Regression Analysis
Testing Deterministic Assumptions
Adjusted 𝑹 𝟐
𝐴𝑑𝑗. 𝑅 2 =1− 𝑒 𝑡 2 (𝑛−𝑘) (𝑦− 𝑦 ) 2 (𝑛−1) ;
The value of the adjusted R2 also ranges from 0 to 1. The higher, the better. It will be lower than the standard 𝑅 2 . It shows the joint explanatory power of all the independent variables to the dependent variable.

33. Simple Linear Regression Analysis
Testing Deterministic Assumptions
RAMSEY Test
This a formal test to check if the model structure is correct or not. Let’s assume a following model,
𝑦 𝑡 = 𝛽 0 + 𝛽 1 𝑥 1𝑡 + 𝛽 2 𝑥 2𝑡 + 𝑒 𝑡
The test equation can be represented as,
𝑦 𝑡 = 𝛽 0 + 𝛽 1 𝑥 1𝑡 + 𝛽 2 𝑥 2𝑡 + 𝛽 3 𝑦 2 + 𝑒 𝑡
where, 𝛽 0 is the constant term, 𝛽 𝑛 are the coefficients, 𝑦 𝑡 is the dependent variable, 𝑥 1𝑡 and 𝑥 2𝑡 are the independent variables and 𝑦 2 is squared estimated values of the dependent variable calculated using the parameter estimates of the main model.
The following null hypothesis is tested,
𝐻 0 ;𝑁𝑜 𝑒𝑟𝑟𝑜𝑟 𝑖𝑛 𝑚𝑜𝑑𝑒𝑙 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒

34. Simple Linear Regression Analysis
Testing Deterministic Assumptions
RAMSEY Test
The following LM or F statistics can used to test the above hypothesis,
𝐿𝑀−𝑠𝑡𝑎𝑡. =𝑛∗ 𝑅 2 ~ 𝜒 2 (𝑔)
Here, n is the number of observations and 𝑅 2 is the coefficient of determination, that explains how much variation in the dependent variable is explained by the independent or explanatory variables.
𝐹−𝑠𝑡𝑎𝑡. = 𝑅𝑆𝑆 𝑅 − 𝑅𝑆𝑆 𝑈𝑅 𝑔 𝑅𝑆𝑆 𝑈𝑅 𝑛−𝑘 ~ 𝐹 (𝑔,𝑛−𝑘)

35. Simple Linear Regression Analysis
Applications in Eviews
Regress a stock returns over market returns

36. Simple Linear Regression Analysis
Testing Stochastic Assumptions
In testing stochastic assumptions of OLS, first of all, the dependent variable should be analyzed for possible structural breaks.
For sharp structural breaks, dummy variables can be included in the model. For smooth structural breaks, non-linear models can be used.
After the structural break tests, the residuals of the estimated model should be checked for heteroskedasticity, autocorrelation and endogeneity issues.

37. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Heteroskedasticity: Testing & Remedies
An important stochastic assumption of the OLS estimation is that the residuals of the estimated model should be homoscedastic meaning the variance of residuals should be constant.
If the variance is not constant and changes overtime, it shows that the residuals are heteroskedastic. If the heteroskedasticity is present, the coefficient estimates will not be efficient and the hypothesis tests will not be reliable.

38. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Heteroskedasticity: Testing & Remedies
White Test
White test is one of the most popular and sensitive tests for testing heteroskedasticity problem in a model.
Let’s assume a following model,
𝑦 𝑡 = 𝛽 0 + 𝛽 1 𝑥 𝑡 + 𝑒 𝑡
Test Equation
𝑒 𝑡 2 = 𝛼 0 + 𝛼 1 𝑥 𝑡 + 𝛼 2 𝑥 𝑡 2 + 𝜀 𝑡
𝐻 0 : 𝛼 1 = 𝛼 2 =0; 𝑁𝑜 𝐻𝑒𝑡𝑒𝑟𝑜𝑠𝑘𝑒𝑑𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦
Test Statistic
𝐿𝑀−𝑠𝑡𝑎𝑡. =𝑛 . 𝑅 2 ∿ 𝜒 2 (𝑔)
where, n is the number of observations, 𝑅 2 is the coefficient of determination and 𝑔 show the number of restrictions.

39. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Heteroskedasticity: Testing & Remedies
White Test
Applications in Eviews

40. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Heteroskedasticity: Testing & Remedies
Breusch – Pagan- Godfrey Test
This test includes just the normal series of the independent variables and exclude the squared series. The test equation is modified as follows,
𝑒 𝑡 2 = 𝛼 0 + 𝛼 1 𝑥 𝑡 + 𝜀 𝑡
𝐻 0 : 𝛼 1 = 𝛼 2 =0; 𝑁𝑜 𝐻𝑒𝑡𝑒𝑟𝑜𝑠𝑘𝑒𝑑𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦
Test Statistic
𝐿𝑀−𝑠𝑡𝑎𝑡. =𝑛 . 𝑅 2 ∿ 𝜒 2 (𝑔)
The null hypothesis and test statistics are same in all the heteroskedasticity test discussed here. The only difference is in the test equations.

41. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Heteroskedasticity: Testing & Remedies
Breusch – Pagan- Godfrey Test
Applications in Eviews

42. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Heteroskedasticity: Testing & Remedies
Harvey Test
It uses logged values of the squared residuals. Also, just like the Breusch-Pagan-Godfrey test, it does not use the squared series of the independent variable.
ln( 𝑒 𝑡 2 )= 𝛼 0 + 𝛼 1 𝑥 𝑡 + 𝜀 𝑡 𝐻 0 : 𝛼 1 = 𝛼 2 =0; 𝑁𝑜 𝐻𝑒𝑡𝑒𝑟𝑜𝑠𝑘𝑒𝑑𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦
Test Statistic
𝐿𝑀−𝑠𝑡𝑎𝑡. =𝑛 . 𝑅 2 ∿ 𝜒 2 (𝑔)

43. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Heteroskedasticity: Testing & Remedies
Harvey Test
Applications in Eviews

44. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Remedies of Heteroskedasticity
It is important to understand the possible reasons for the heteroskedasticity problem. One of the reasons might be the nature of the data used. For example, the models that use sectoral, households or cross-sectional data are more likely to face this problem.
The other reason might be the model specification. An important independent variable might have been omitted. In the test equation output of the heteroskedasticity tests discussed above, check for the variables that have significant effect on the residuals and think of modifying the model.

45. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Remedies of Heteroskedasticity
Econometrics offers following remedies for the heteroskedasticity problems.
White Method
This method is one of the most popular methods to correct heteroskedasticity issues. Instead of modifying the model, it corrects the standard error to make them consistent.
Applications

46. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Autocorrelation: Testing & Remedies
Autocorrelation problem arises when the residuals are correlated with each other.
Testing Autocorrelation
Breusch-Godfrey LM Test
Breusch-Godfrey serial correlation LM test is one of the most popular test for test autocorrelation problem in the residuals. Let’s assume a following model,
𝑦 𝑡 = 𝛽 0 + 𝛽 1 𝑥 𝑡 + 𝑒 𝑡
Test Equation
𝑒 𝑡 = 𝛼 0 + 𝛼 1 𝑥 𝑡 + 𝛾 1 𝑒 𝑡−1 + 𝛾 2 𝑒 𝑡−2 +…+ 𝛾 𝑝 𝑒 𝑡−𝑝 + 𝜀 𝑡

47. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Autocorrelation: Testing & Remedies
Autocorrelation problem arises when the residuals are correlated with each other.
Testing Autocorrelation
Breusch-Godfrey LM Test
𝐻 0 : 𝛾 1 = 𝛾 2 =…= 𝛾 𝑝 =0; 𝑁𝑜 𝑎𝑢𝑡𝑜𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑝𝑟𝑜𝑏𝑙𝑒𝑚
𝐻 1 : 𝛾 𝑝 ≠0;𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑓𝑜𝑟 𝑜𝑛𝑒 𝐴𝑢𝑡𝑜𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑝𝑟𝑜𝑏𝑙𝑒𝑚 𝑒𝑥𝑖𝑠𝑡𝑠
Test Statistic
𝐿𝑀−𝑠𝑡𝑎𝑡. =𝑛 . 𝑅 2 ∿ 𝜒 2 (𝑔)
where, n is the number of observations, 𝑅 2 is the coefficient of determination and 𝑔 show the number of restrictions.
Applications

48. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Autocorrelation: Testing & Remedies
Testing Autocorrelation
Correlogram
This is another popular way to test the presence of autocorrelation among the residuals. It calculates the correlation between the consecutive residuals as follows,
𝜌 𝑖 = 𝐶𝑜𝑣( 𝑒 𝑡 , 𝑒 𝑡−1 ) 𝜎 𝑒 𝑡 , 𝜎 𝑒 𝑡− where, 𝜌 𝑖 (rho) is the correlation coefficient, 𝐶𝑜𝑣( 𝑒 𝑡 , 𝑒 𝑡−1 ) is the covariance between the residuals and 𝜎 𝑒 𝑡 is standard deviation of the residuals.
The null hypothesis of no autocorrelation is tested using the following test statistic,
𝑄−𝑠𝑡𝑎𝑡. =𝑛 𝑖=1 𝑝 𝜌 𝑖

49. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Autocorrelation: Testing & Remedies
Testing Autocorrelation
Correlogram
where, n is the number of observations and 𝜌 𝑖 is the correlation coefficient.
For example, if the correlation coefficients between 𝑒 𝑡 & 𝑒 𝑡−1 and 𝑒 𝑡−1 , 𝑒 𝑡−2 are 0.70 and 0.98 respectively, then the Q-stat. would be calculated as,
𝑄−𝑠𝑡𝑎𝑡. 𝑓𝑜𝑟 𝐹𝑖𝑟𝑠𝑡 𝑙𝑎𝑔=𝑛∗0.70
𝑄−𝑠𝑡𝑎𝑡. 𝑓𝑜𝑟 𝑆𝑒𝑐𝑜𝑛𝑑 𝑙𝑎𝑔=𝑛∗( )
Applications

50. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Remedies for Autocorrelation
The autocorrelation problem may arise due to data related issues. It is common in time series data or in a model having one or more non-stationary variables.
Data manipulation might also cause this problem. For example, if the original yearly data is converted into quarterly data manually, the autocorrelation in the residuals may result.
Model related issues may also result in the autocorrelation problem. The absence of a necessary variable in the model may cause this problem.
Another possible reason is the presence of seasonality in the data.

51. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Remedies for Autocorrelation
Following are the possible ways to correct the autocorrelation in the residuals,
Add new variables in the model
Add lagged series of the dependent variable
Take seasonality into account by adding dummy variable
HAC method

52. Simple Linear Regression Analysis
Testing Stochastic Assumptions
Remedies for Autocorrelation
HAC Method
HAC method, also known as Newey-West method, is commonly used to correct both autocorrelation and heteroskedasticity problems.
It takes into account the lagged values of the dependent variable and correct standard errors. However, the coefficient estimates remain the same.
Applications

53. Multiple Linear Regression Analysis
Applications

54. Unit Root Tests
The main purpose of the unit root tests is to check if a series is stationary or non- stationary.
Tests if the shocks in a series is permanent or temporary.
If a series is non-stationary or has unit root, it means the shocks last permanently.
If it is stationary or has no unit root, it means shocks disappear with time and series comes to its mean form. This is also sometimes referred to as mean reverting process.
Unit root in an important economic variable may be devastating for an economy. Suitable policies should be adopted to absorb the shocks.

55. Unit Root Tests DF (Dickey-Fuller) Test Strategy
In this strategy, difference series of Yt is regressed on lagged values of original (level) series and the coefficient of Yt-1 is tested for unit root.
∆ 𝑦 𝑡 =𝛼 𝑦 𝑡−1 + 𝜀 𝑡
The null hypothesis is:
𝐻 0 : 𝛼 = 0; Series is non-stationary (unit root)
𝐻 1 : 𝛼 < 0; Series is stationary (no unit root)

56. Unit Root Tests DF (Dickey-Fuller) / ADF (Augmented DF) Unit Root Test
The test statistic as ‘tau-statistic’.
𝑇𝑎𝑢 𝑇= 𝛼 𝛿𝑡 𝛼 ; by Dickey and Fuller (1979)
In ADF approach, lagged valued of the dependent variable is added into the test equation. Compare the following DF and ADF test equations.
∆ 𝑦 𝑡 = 𝑍 𝑡 𝛿+𝛼 𝑦 𝑡−1 + 𝜀 𝑡 ; DF test equation ∆ 𝑦 𝑡 = 𝑍 𝑡 𝛿+𝛼 𝑦 𝑡−1 + 𝑗=1 𝑡 𝛽 𝑗 ∆ 𝑌 𝑡−𝑗 +𝜀 𝑡 ; ADF test equation

57. Unit Root Tests DF (Dickey-Fuller) / ADF (Augmented DF) Unit Root Test
Applications

58. Thank You