2. Regression Analysis

Learning Objectives

• Define regression analysis and explain the role of the dependent and independent variables
• Distinguish between simple linear regression and multiple linear regression
• Calculate and interpret OLS-estimated coefficients using Indian economic examples
• Explain what R-squared, t-statistics, and p-values mean in a regression output
• Identify when to use non-linear regression and give an Indian example
• List the key assumptions of OLS and explain what goes wrong when each is violated
• Apply regression thinking to policy-relevant questions about India's economy

Quick Answer

Regression analysis is a statistical method for estimating the relationship between one outcome variable and one or more explanatory variables. The most common form, ordinary least squares (OLS), fits a line through data points by minimising the total squared distance between actual and predicted values. In Indian economics, regression is used to answer questions like "how much does a one-unit increase in the literacy rate raise state GDP per capita?" or "what is the effect of the RBI's repo rate on the inflation rate?" Understanding regression means being able to both run such analyses and critically interpret the results reported in research papers and government reports.

Introduction

Regression analysis is the workhorse of applied econometrics. Virtually every empirical economics paper — whether published by the RBI, NITI Aayog, or in an academic journal — uses some form of regression. The goal is not just to describe data but to estimate a quantitative relationship: by how much does Y change when X changes by one unit, holding everything else constant?

In India's policy context, this "holding everything else constant" feature is what makes regression so useful. Inflation depends on many things simultaneously — money supply, oil prices, supply shocks, exchange rates. Regression allows analysts to isolate the separate contribution of each factor.

What is Regression Analysis?

Regression analysis establishes a relationship between a dependent variable (the thing you want to explain) and one or more independent variables (the factors that explain it). The basic linear regression model is:

Y = α + βX + ε

Where:

• Y = dependent variable (e.g., inflation rate)
• X = independent variable (e.g., money supply growth)
• α = intercept (value of Y when X = 0)
• β = slope coefficient (how much Y changes when X increases by 1 unit)
• ε = error term (captures all other factors not included in the model)

The coefficient β is what we are most interested in estimating.

Types of Regression

Simple Linear Regression

Simple linear regression examines the relationship between one dependent variable and one independent variable. The OLS method estimates α and β by minimising the sum of squared residuals.

Indian example: Suppose you want to know how average life expectancy across Indian states relates to per capita health spending. Your regression would be:

Life Expectancy = α + β (Health Spending per capita) + ε

If β = 0.8, it means every additional ₹1,000 per capita in health spending is associated with 0.8 more years of life expectancy, on average across states, holding nothing else constant (because nothing else is in this model).

Multiple Linear Regression

Multiple linear regression extends the simple model to include several independent variables simultaneously:

Y = α + β₁X₁ + β₂X₂ + ... + βₙXₙ + ε

This is more realistic because most economic outcomes have multiple causes.

Indian example: Modelling inflation in India:

Inflation = α + β₁(Repo Rate) + β₂(Oil Price) + β₃(Money Supply Growth) + ε

Each β now represents the partial effect of that variable — the effect of, say, the repo rate on inflation, holding oil prices and money supply growth constant. This is exactly the kind of model the RBI uses to understand how its interest rate tool affects prices.

Non-linear Regression

When the relationship between variables is not linear — for instance, when returns to scale are diminishing — non-linear regression models are appropriate.

Indian example: The relationship between population density and urbanisation in India is not strictly linear. Initially, rising density accelerates urbanisation as agglomeration economies kick in. Beyond a threshold, congestion and slum formation slow the relationship. A logarithmic or quadratic model captures this better than a straight line.

Real-world Examples in India

Example 1: Impact of Education on Economic Growth

India has invested heavily in education through programmes like Sarva Shiksha Abhiyan and the National Education Policy. Regression analysis can quantify the payoff.

Model:

• Dependent variable: State-level GDP growth rate (%)
• Independent variables: State literacy rate (%), per capita education spending (₹), urban share (%)
• Method: Multiple OLS regression using NITI Aayog state data

Interpretation: If the estimated coefficient on literacy rate is 0.15, it means — after controlling for spending and urbanisation — a one percentage point rise in the literacy rate is associated with 0.15 percentage points higher annual GDP growth. Policymakers can compare this to the cost of achieving that literacy gain.

Example 2: Relationship Between Inflation and Interest Rates

The RBI's primary tool for controlling inflation is the repo rate. Regression analysis tests whether this tool actually works.

Model:

• Dependent variable: CPI inflation rate (%)
• Independent variable: Repo rate set by the RBI (%)
• Method: Simple OLS using monthly RBI data

Expected result: A negative β — higher repo rates are associated with lower inflation — consistent with monetary theory. The size of β tells us the effectiveness of the interest rate channel.

Important caveat: With monthly data, inflation today may depend on interest rates from six to twelve months ago (monetary policy lags). A more careful model would use lagged values of the repo rate.

Example 3: Urbanisation and Infrastructure Development

As India urbanises rapidly, researchers use regression to understand what drives the pace of urban growth.

Model:

• Dependent variable: Annual rate of urbanisation across districts (%)
• Independent variables: Per capita infrastructure investment (₹), initial population density, distance to nearest railway station
• Method: Non-linear (log-linear) OLS

Indian context: Districts in the Indo-Gangetic plain show different urbanisation dynamics from coastal districts; including regional dummies or estimating separate regressions by zone is standard practice.

Understanding Regression Output

A typical regression output table looks like this (illustrative):

Variable	Coefficient (β)	Standard Error	t-statistic	p-value
Intercept (α)	2.10	0.45	4.67	0.000
Repo Rate	−0.38	0.12	−3.17	0.002
Oil Price (USD/barrel)	0.09	0.03	3.00	0.004
Money Supply Growth	0.22	0.08	2.75	0.008
R-squared	0.71

Reading this output:

• The repo rate coefficient (−0.38) means a 1 percentage point rise in the repo rate reduces inflation by 0.38 percentage points, holding oil price and money supply constant
• The p-value of 0.002 means there is only a 0.2% chance of observing this coefficient if the true effect were actually zero — it is statistically significant
• R-squared of 0.71 means this model explains 71% of the variation in inflation across the sample

Key OLS Assumptions

OLS gives the Best Linear Unbiased Estimator (BLUE) only when these assumptions hold:

1. Linearity: The relationship between Y and Xs is linear (or correctly specified)
2. No perfect multicollinearity: Independent variables are not perfectly correlated with each other
3. Zero conditional mean error: E(ε|X) = 0 — no systematic error; no important variable is left out
4. Homoscedasticity: Variance of errors is constant across observations
5. No autocorrelation: Errors are not correlated with each other (especially important with time series)
6. Normality of errors: For small samples; needed for exact inference

When these fail, OLS estimates can be biased, inefficient, or both. This is why checking diagnostic tests is as important as running the regression itself.

Conclusion

Regression analysis is indispensable for quantitative economic reasoning. Mastering it means understanding not just the mechanics of OLS but also the economic logic that determines what variables to include, the statistical diagnostics that verify your model is trustworthy, and the careful language needed when translating coefficients into policy claims. Every RBI monetary policy report, every NITI Aayog growth study, and every serious economics research paper rests on regression at its core.

Key Terms

Term	Definition	Related Concept
Regression analysis	A statistical method for estimating relationships between a dependent variable and one or more independent variables	OLS, econometrics
OLS (Ordinary Least Squares)	Estimation method that minimises the sum of squared residuals	Regression, BLUE estimator
Coefficient (β)	The estimated change in Y for a one-unit change in X, holding other variables constant	Interpretation, partial effect
R-squared	Proportion of variation in the dependent variable explained by the model (0 to 1)	Goodness of fit
t-statistic	Ratio of the coefficient to its standard error; used to test statistical significance	Hypothesis test, p-value
p-value	Probability of observing this coefficient if the true effect were zero; low p = significant	Statistical significance
Multicollinearity	High correlation between independent variables making it hard to isolate individual effects	OLS assumptions, VIF
Heteroscedasticity	Non-constant variance of error terms, violating an OLS assumption	Robust standard errors
Autocorrelation	Correlation of error terms across time periods, common in time series	Durbin-Watson test, ARIMA
Omitted variable bias	Bias in coefficients when a relevant variable is excluded from the model	Specification error
Partial effect	The effect of one independent variable on Y, holding all other variables constant	Multiple regression
Residual	The difference between the actual value of Y and the value predicted by the regression	OLS, goodness of fit

Common Mistakes

Misconception: A regression coefficient tells you the total effect of X on Y. Why it's wrong: In a multiple regression, each coefficient is a partial effect — the effect of that variable holding all other included variables constant. If you omit an important variable, the coefficient absorbs the influence of the missing variable, biasing the estimate. Correct understanding: A coefficient only represents the isolated effect of that variable within the specific model you estimated. Add or remove variables and the coefficients change. Always state what is being controlled for.

Misconception: A p-value below 0.05 means the result is economically important. Why it's wrong: Statistical significance and economic significance are different. With a very large dataset (India's PLFS has hundreds of thousands of observations), even a tiny, economically trivial effect (e.g., education raises income by ₹2 per month) will be statistically significant. Conversely, a large dataset is not a guarantee of finding the effect you expect. Correct understanding: Always report the magnitude of the coefficient alongside its significance. Ask: is this coefficient large enough to matter for real decisions? In Indian policy contexts, a statistically significant but tiny effect rarely justifies a costly intervention.

Misconception: If R-squared is high, the model is correctly specified. Why it's wrong: Two trending variables will have a high R-squared even if they are completely unrelated — the spurious regression problem. Also, a model can have a moderate R-squared and still be correctly specified if the outcome variable is genuinely difficult to predict (like daily stock returns). Correct understanding: Test for stationarity before running regressions on time series. Run diagnostic tests for autocorrelation (Durbin-Watson) and heteroscedasticity (Breusch-Pagan). R-squared is a measure of fit, not correctness.

Comparison and Connections

Feature	Simple Regression	Multiple Regression	Non-linear Regression
Number of X variables	One	Two or more	One or more
Equation form	Y = α + βX + ε	Y = α + β₁X₁ + β₂X₂ + ε	Y = f(X) + ε (e.g., log, quadratic)
When to use	One dominant explanatory factor	Multiple causes of Y	Diminishing returns, threshold effects
Indian example	Repo rate → inflation	Literacy + spending → GDP growth	Population density → urbanisation
Risk	Omitted variable bias	Multicollinearity	Model misspecification
Typical Indian data source	RBI monthly repo and CPI data	NITI Aayog state panel data	Census district-level data

Practice Questions

Recall

1. Write out the simple linear regression equation and identify what each term represents. Answer guidance: Y = α + βX + ε; Y is the dependent variable, X is the independent variable, α is the intercept, β is the slope coefficient, ε is the error term.

2. What does a p-value of 0.03 on a regression coefficient mean? Answer guidance: There is a 3% probability of observing a coefficient this large (or larger in magnitude) purely by chance if the true population coefficient were zero. Since 0.03 < 0.05, it is conventionally considered statistically significant.

Understanding 3. Explain why a researcher studying the impact of infrastructure investment on state GDP growth would prefer multiple regression over simple regression. Answer guidance: States differ in many ways — literacy, urbanisation, governance quality. Without controlling for these, the estimated coefficient on infrastructure absorbs their influence, biasing the result. Multiple regression holds these constant.

4. What is the difference between statistical significance and economic significance? Give an Indian example. Answer guidance: Statistical significance says the effect is unlikely to be zero. Economic significance says the effect is large enough to matter. Example: a study might find that a free mid-day meal programme raises school attendance by 0.05 percentage points (statistically significant in a large sample) but this is too small to justify the programme cost — it lacks economic significance.

Application 5. A researcher estimates: State GDP per capita = 12,500 + 85 × (Literacy Rate). Interpret the coefficient on literacy rate. What limitation should be mentioned? Answer guidance: Each one percentage point rise in the literacy rate is associated with ₹85 higher GDP per capita across states. Limitation: this is a simple regression; the literacy rate is correlated with other factors (urbanisation, infrastructure) that also raise GDP, so this coefficient likely overstates the pure literacy effect.

6. You run a regression of India's monthly exports on the USD/INR exchange rate and get R-squared = 0.78. A classmate says "this proves the exchange rate explains exports." What would you say? Answer guidance: R-squared of 0.78 shows the exchange rate explains 78% of variation in exports in this sample. But we need to check: (a) are both series stationary? (b) what other variables are omitted? (c) is the direction of causation clear? A high R-squared does not prove causation or correct specification.

Analysis 7. A study of 28 Indian states finds that states with higher FDI inflows have higher manufacturing output. List three alternative explanations (other than FDI causing manufacturing growth) that would need to be ruled out. Answer guidance: (1) Reverse causality — high manufacturing output attracts FDI. (2) Third variable — coastal states have both high FDI and high manufacturing due to port access. (3) Policy endogeneity — states that already have better infrastructure and governance attract both FDI and manufacturing.

8. The RBI estimates that a 1 percentage point increase in the repo rate reduces inflation by 0.4 percentage points. A politician argues this means cutting the repo rate by 2 points will raise inflation by 0.8 points. What regression-based objection would an economist raise? Answer guidance: Regression estimates are averages over the sample period and linear approximations. Effects may not be symmetric (rate cuts vs. rises). The relationship may have changed post-COVID. The estimate holds other things constant — but in practice, cutting rates also affects growth, rupee depreciation, and oil import costs, which in turn affect inflation through other channels.

FAQ

Why do we use OLS and not some other estimation method? OLS is preferred because, under the Gauss-Markov assumptions, it produces the Best Linear Unbiased Estimator — no other linear estimator has smaller variance. It is also computationally simple and its results are easy to interpret. When Gauss-Markov assumptions fail (e.g., heteroscedasticity, endogeneity), economists switch to alternatives like Weighted Least Squares, Instrumental Variables, or Generalised Method of Moments — but OLS remains the starting point and the benchmark.

How do I choose which variables to include in a regression? The primary guide is economic theory. Include variables that the theory says should matter; exclude variables for which there is no theoretical justification, as adding too many variables can cause overfitting. In practice, researchers also consult previous literature, run sensitivity checks (do results hold when I add or remove this variable?), and use information criteria like AIC or BIC to compare model fit. For Indian state-level regressions, standard controls include initial income per capita, literacy rate, urbanisation rate, and infrastructure index.

What is endogeneity and why does it matter for Indian policy research? Endogeneity means that an independent variable is correlated with the error term, violating OLS assumption 3. The classic case in Indian research is studying the effect of government spending on growth: states that are growing faster generate more tax revenue, which funds more spending, so the causation runs both ways. This makes the OLS estimate biased. Instrumental Variables (IV) is the standard fix — find a variable that affects spending but does not directly affect growth through any other channel. In Indian studies, political representation, historical colonial infrastructure, or rainfall (for agricultural studies) are commonly used instruments.

What does it mean for a coefficient to be negative? A negative coefficient means that when X increases, Y decreases, on average, holding other variables constant. For example, if the coefficient on the repo rate in an inflation regression is −0.38, it means higher interest rates are associated with lower inflation — exactly what monetary theory predicts. A negative coefficient is not bad; it simply indicates the direction of the relationship.

Can regression be used with categorical or qualitative variables? Yes. Categorical variables (state identity, policy regime, gender) are entered as dummy variables — coded 0 or 1. For example, if studying whether being a coastal state (1) or non-coastal (0) affects export intensity, the dummy variable's coefficient tells you the average difference in export intensity between coastal and non-coastal states, holding other factors constant. This technique is widely used in Indian cross-state studies. When the dependent variable itself is categorical (e.g., above-poverty versus below-poverty), logit or probit regression is used instead of OLS.

Quick Revision

• Regression estimates the relationship between a dependent variable Y and one or more independent variables X
• Simple regression: one X variable; multiple regression: two or more X variables simultaneously
• OLS minimises the sum of squared residuals to estimate α and β
• The coefficient β is the partial effect — change in Y per unit change in X, holding other variables constant
• R-squared (0 to 1) measures goodness of fit, not correctness of the model
• t-statistic = coefficient / standard error; p-value < 0.05 conventionally means statistically significant
• OLS is BLUE under the five Gauss-Markov assumptions: linearity, no perfect multicollinearity, zero mean error, homoscedasticity, no autocorrelation
• Omitted variable bias: leaving out a relevant variable that is correlated with X biases the coefficient
• Heteroscedasticity and autocorrelation do not bias coefficients but make standard errors incorrect
• Statistical significance ≠ economic significance — always report the size of the effect, not just its p-value
• Spurious regression: two trending series show high R-squared even without a meaningful relationship — always test for stationarity first
• Key Indian applications: RBI inflation modelling, NITI Aayog growth projections, state-level panel studies of education and health outcomes

Related Topics

Prerequisites: Basic statistics (mean, variance, correlation coefficient), introductory econometrics (OLS concept), macroeconomics (GDP, inflation, monetary policy)

Related Topics: Time series analysis and stationarity testing, index numbers and data interpretation, RBI monetary policy and inflation targeting

Next Topics: Time series analysis, ARIMA and VAR models, panel data econometrics