Assignment Code: BECE-142/AST/TMA/2022-23

Course Code: BECE-142

Assignment Name: Applied Econometrics

Year: 2022-2023

Verification Status: Verified by Professor

Maximum Marks: 100

Answer all the questions

A. Long Answer Questions (word limit-500 words) 2 × 20 = 40 marks

Q1) Explain, theoretically, the consequences of omitting relevant variables in econometric modelling.

Ans) Quite sometimes, we may unintentionally leave out a crucial explanatory variable from the regression model. Consider a regression model in which the dependent variable Y is actually related to two variables X2 and X3 ,like:

Y= β_1+β_2 X_2+β_3 X_3+u_i Equation (1)

Not knowing this, suppose we wrongly construe and specify the model as:

Y_i= β_1+β_2 X_2i+u_i Equation (2)

We know that the estimate of β_2, β ̂_2 or the OLS estimator for β_2, can be obtained from Equation (2) as:

Other similar consequences of omitting a variable can be stated as the following.

The estimates of the regression coefficients will be skewed if a variable that is correlated with the dependent variable and either of the explanatory factors is omitted. The nature of the correlations between (a) the dependent variable and omitted variables and (b) independent variables and omitted variables determines the bias's nature. The nature of bias can be summed up as follows if the real regression model is as in (1), where X3 is the omitted variable:

The OLS estimates will also be inconsistent to the degree that the estimates will still be biased even for huge samples.

If X2 and X3 are not related in the sample, the value of ∂ ̂_2 will be zero and the estimate for slope coefficient will be unbiased as well as consistent. However, the estimate for intercept would remain biased, unless the mean of X3 is zero.

The error variance estimated from the mis-specified model will also be a biased estimator of the true error variance σ2 .Consequently, the estimates for the variance of slope coefficient will also be biased and the variance of the estimated slope coefficient β ̂_2of the mis-specified model will be overestimated.

All of them together will substantially impair the reliability of the confidence interval and the results of the hypothesis testing.

Q2) Discuss the effect of lags on ‘market equilibrium’ with suitable examples.

Ans) The example of the effect of lags on ‘market equilibrium’ is as follows:

Assume that the supply and demand equations for a single good on the market for time t are as follows:

These oscillations will either converge to or diverge from (P ̅ ). We often plot quantity along the horizontal axis and price along the vertical axis in demand-supply curve diagrams. Therefore, the supply and demand curves' slopes would equal the reciprocals of the supply and demand functions' slopes (i.e., (β_1 and β_2 ). There are three conceivable results, namely convergent, divergent, and neither convergent nor divergent, even if the supply and demand functions have their normal slopes (i.e., positive, and negative, respectively). If the demand curve is flatter than the supply curve, the convergence solution is reached. In the reverse scenario, where the supply curve is flatter than the demand curve, the divergent solution will manifest. When the demand and supply curves are both equally steep, this is a particular example. In such cases, |β_1/β_2 |=1.

Now, suppose that initially the supply of the good concerned has fallen short of the equilibrium amount because of some disturbance like drought or flood. Let the initial supply be Q1 as shown in the figure below. This amount would be demanded in the initial period if the price is P_1. At P=P_1, the consumers demand is not matched with the producer’s supply. As a result, in period 2, the price will sharply drop from P_1 to P_2.

In the Cobweb model, this period’s price influences the next period’s supply. Hence, the price P_1 of the initial period (period 1) determines the supply of period 2. In other words, the price P_1 induces the entrepreneurs to supply more output in period 2. This way, the process continues indefinitely producing a Cobweb pattern. The fluctuation in prices will be higher than the equilibrium price P ̅ in one period and lower than P ̅ in the next period. It however converges to the equilibrium level at the point of intersection of demand and supply curves

.

B. Medium Answer Questions (word limit-250 words) 3 × 10 = 30 marks

Q3) Outline how R2 and adjusted-R2 serve as indicators of ‘goodness of fit’ of a regression model.

Ans) R2 and adjusted-R2 as indicators of ‘goodness of fit’ of a regression model:

The coefficient of determination (R2) is one of the measures of goodness of fit of a regression model. Recall that it is defined as:

Q4) Indicate the form of a Logit Model. Specify why the OLS method of estimation cannot be applied here.

Ans) Let us add the error term in the equation, we get:

Thus, the values of ln (p_i/(1-p_i )) and ln(1-p_i ) are estimated as (β_0+β_1 x_i ) and -ln(1+e^((β_0+β_1 x_i ) ) ) respectively. The objective is to maximise the LLF i.e., Equation (3) with respect to β_0 and β_1 using the values of x which are known. The resulting solutions become non-linear in the parameters because of the presence of e^(β_0+β_1 x_ ) term. This is the reason why the OLS method cannot be applied for the Logit model. Once the values of the parameters are known, we can easily estimate the logistic equation.

Q5) Show that ‘exact identification’ is a sufficient condition for the unique determination of a system of equations.

Ans) Let us consider the earlier supply function but with a new demand function where we assume that the quantity demanded is determined by both ‘price and income (I)’. The demand and supply functions are:

C. Short Answer Questions (word limit 100 words) 2 × 3 × 5 = 30 marks

Q6) Differentiate between:

(a) Quantitative Research and Qualitative Research.

Ans) The differences between quantitative research and qualitative research are as follows:

(b) In-sample forecast and out-of-sample forecast.

Ans) The differences between in-sample forecast and out-of-sample forecast are as follows:

An in-sample prediction forecasts values for the estimation period and compares them to the actual results using a subset of the dataset. To put it another way, in-sample forecasting evaluates how well the selected model matches the data in a particular sample. In order to estimate the future value of the regressand, an out-of-sample forecasting uses all the values in the sample's accessible data.

By dividing a given data set into an in-sample period for initial parameter estimate and model selection and an out-of-sample period for assessing forecasting performance, statistical tests of a model's prediction ability are frequently carried out.

Evidence based on in-sample performance, which might be more susceptible to outliers and data mining, is generally seen as less reliable than evidence based on out-of-sample forecast performance. Out-of-sample forecasts also more accurately represent the data that the forecaster has access to in "real time."

(c) Autoregressive Model and Autoregressive Distributed Lag Model.

Ans) The differences between Autoregressive Model and Autoregressive Distributed Lag Model are as follows:

Regressing a value from a time series on earlier values from the same time series is known as an autoregressive model of regression. The dependent variable from the prior time period serves as the predictor in these models.

Using a regression equation, distributed lag models predict the current values of a dependent variable based on both the current values of an explanatory variable and the lagged values of this explanatory variable.

The models are referred to as "distributed lag models" if the lags are present solely in the exogenous variables, with both their current and lagged values appearing as exogenous variables. The model is referred to as an "autoregressive model" if the lags are present solely in the endogenous variables and they act as exogenous regressors. The term "autoregressive distributed lag models" refers to models where the set of explanatory variables includes lags from both exogenous and endogenous factors.

Q7) Write short notes on the following.

(a) Instrumental Variables (IV) Method.

Ans) To estimate the causal linkages, instrumental variables are used. When controlled studies are not possible or when a therapy cannot be successfully administered to every unit, it is especially useful. Therefore, it makes sense that the approach of IVs is helpful when a relevant explanatory variable is linked with the error term [since in this situation, ordinary least squares is known to produce estimates that are biased].

An instrument is considered to be legitimate if it causes changes in the explanatory variable but has no independent impact on the dependent variable. This enables a researcher to identify the causal relationship between the explanatory and dependent variables. The IV technique uses a single equation. This indicates that each equation in an equation system is treated separately. It was created as a remedy for the "simultaneous equation bias" and is suitable for systems that have been overidentified.

(b) Pooled Cross Section Data.

Ans) The process of obtaining pooled cross sectional data involves independently obtaining random samples from a large population at several points in time. Both cross-sectional and time series elements are included in panel data sets (it consists of time series data for each statistical unit in the cross section). Take two cross-sectional home surveys conducted in 1985 and 1990, for instance. A random sample of homes was questioned in 1985 about things like income, savings, family size, etc. Using the same survey questions, a new random sample of households was collected in 1990. By merging the two years, we may create a pooled cross section to expand our sample size. Analysis of the consequences of a new government policy can be effectively done by combining cross sections from various years. Data from the years before and after a significant policy change will be gathered.

(c) Linear Static Panel Data Model.

Ans) Suppose for each cross section unit we collect data on same set of variables for T time periods. Let X be a vector of k exogenous variables which affect Y. At any time, point ‘t’, the population model is like:

Yit = Xit β + ci + uit , t = 1,2,….,T, I = 1,2,…,N

where ci is the unobserved effect and it u is the random error term. The aforementioned equation is a panel regression with the variables I and "t" standing in for the cross section unit and time, respectively. The "ordinary least square" method is the one that is most frequently used to estimate the parameters (OLS). The exogenous nature of the explanatory variables and their lack of correlation with the random error term are presumptions made by the OLS. The main purpose of panel data is to address the issue of omitted variables. We take time into account in panel data models to account for the unobserved effect (like quality in the example considered above). We presumptively consider the "unobserved effects" to be random variables. An example of a linear static panel data model is this. Because all of the explanatory variables are contemporaneous dates that correspond to the value of Y in period t, the model is static. In contrast, a dynamic panel data model permits the inclusion of one or more lag dependent variables as a "partial adjustment mechanism" in the models.