Step 1: What Is Econometrics?
- Econometrics is the use of statistical methods to measure economic relationships, test economic theories and forecast economic outcomes, mostly with data that come from people’s own choices rather than from experiments.
- Every empirical study follows the same seven steps: question, economic model, econometric model, data, estimation, testing, interpretation.
- An econometric model has parameters (the \( \beta \)s), variables, and an error term \( u \) that holds everything left out. Whether anything in \( u \) is related to the explanatory variables decides whether a result can be read as causal.
- Data come as cross-sections, time series, pooled cross-sections and panels, and the type decides the method.
- In 1,000 workers, one more year of school goes with a wage about 10.3% higher (\( R^2 = 0.139 \)). That is a correlation; ability, family background and selection all stand between it and a causal effect.
What econometrics is for
Economic theory tells you the direction of many effects. More education should raise earnings. A higher price should reduce the quantity demanded. A rise in interest rates should slow inflation. What theory almost never tells you is how much. Is a year of school worth 2% more pay or 20%? Does a 10% rise in fuel tax cut petrol use by 1% or by 8%? Governments, central banks and firms need those numbers before they act. Econometrics is the toolkit that produces them from real data.
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The word joins economics and metrics, measurement. Statistics supplies the tools: means, variances, sampling distributions and tests. What makes econometrics a subject of its own is the kind of data economists have. A chemist can heat one beaker and leave another alone. An economist usually cannot give one group of people more schooling and deny it to another just to see what happens. Most economic data come from people, firms and governments making their own choices, and those choices are tangled together. Much of econometrics is about untangling them.
Econometricians do three broad jobs, and one tool, regression analysis, sits at the centre of all three.
| Job | The question it answers | Example |
|---|---|---|
| Estimation | How big is the effect? | How much does an extra year of education raise hourly pay? |
| Inference | Do the data support the theory? | Do higher minimum wages reduce employment in fast-food restaurants? |
| Forecasting | What will happen next? | What will inflation be next quarter? What will a 2,000-square-foot house sell for? |
You will meet each of these again and again in this guide. Steps 3 to 10 build the regression model and show how to estimate and test it. Steps 11 to 16 deal with the problems that stop a regression measuring what you want. Steps 17 to 19 turn to forecasting, and Step 20 brings it all together in an empirical project.
The seven steps of an empirical study
Every empirical study, from a two-page coursework project to a central-bank research paper, moves through the same stages. If you are writing a dissertation or an IB Extended Essay, these stages are also a good outline for your methods chapter.
| Step | What you do |
|---|---|
| 1. Question | Ask a precise question. “Does education matter?” is too vague. “By what percentage does one extra year of schooling raise the hourly wage of adult workers, holding experience fixed?” can be answered with data. |
| 2. Economic model | Use theory to say which variables matter. Human-capital theory says the wage depends on skills built through education and experience: \( \text{wage} = f(\text{educ}, \text{exper}, \text{other factors}) \). |
| 3. Econometric model | Choose a functional form, give the unknown numbers symbols, and add an error term for everything left out. |
| 4. Data | Find a dataset that measures the variables, check it for errors and missing values, and describe it before running anything. |
| 5. Estimation | Turn the data into numbers \( \hat\beta_0, \hat\beta_1, \dots \) with a method such as ordinary least squares (OLS, Step 3). |
| 6. Testing and checking | Ask whether the estimates are statistically different from zero and whether the method’s assumptions hold. If the checks fail, go back to step 3 and revise the model. |
| 7. Interpretation | Translate the numbers back into an answer to the question, in plain words, with an honest statement of how confident you are and what could be wrong. |
Economic models and econometric models
An economic model names the variables but does not say how they combine or what the numbers are. To estimate anything, you must turn it into an econometric model: an equation with a definite form, unknown parameters and an error term. For the wage question, a standard choice is
Each part has a name you will use for the rest of the guide.
You never see the parameters. You see data, and from the data you compute estimates. The whole of econometric theory is about how close the estimates are likely to be to the parameters, and when they are close at all.
The error term
No two workers with the same schooling and experience earn exactly the same wage. Some are more able, some more motivated, some luckier, some live in high-wage cities. All of these influences, plus any measurement error in the wage itself, go into the error term \( u \) (also called the disturbance).
The error term is not a nuisance to be ignored. It is where most of the danger in econometrics lives. If something inside \( u \) is related to an explanatory variable, the estimate of that variable’s effect will be wrong, however much data you have. Get into the habit of asking one question about every model you meet: what is in the error term, and is it related to the \( x \) variables?
Ceteris paribus: how to read a parameter
The parameter \( \beta_1 \) measures the effect of one more year of education on \( \log(\text{wage}) \) holding experience and everything in \( u \) fixed. Economists call this the ceteris paribus effect, Latin for “other things being equal”. Because the dependent variable is in logs, \( 100\beta_1 \) is approximately the percentage change in the wage for one more year of school. If \( \beta_1 = 0.08 \), an extra year raises the wage by about 8%, other things equal. Step 8 explains exactly why logs work this way.
A student wants to know how a house’s size, age and distance from the city centre affect its price. Write an econometric model, state the expected signs and say what the error term contains.
The four types of economic data
The structure of a dataset decides which methods work and which problems you will face. Before you run anything, ask two questions: how many units are there, and how many periods does each unit appear in?
| Type | Units and periods | Example in the companion data | Main issue |
|---|---|---|---|
| Cross-section | Many units, one period | wages.csv: 1,000 workers; houses.csv: 500 house sales | Row order does not matter; random samples give independent observations |
| Time series | One unit, many periods | us_macro.csv: US quarterly data, 1959–2009 (203 quarters) | Observations depend on the past; trends and seasons (Steps 17–19) |
| Pooled cross-section | Different random samples from different periods | A household survey in 2010 and again in 2020 with new households | Lets you see how a relationship changed over time |
| Panel | The same units, several periods | countries_panel.csv: 60 countries, 2000–2019 (1,200 rows) | Compare each unit with itself over time (Step 15) |
Figure 2 shows why time series need their own methods. Unemployment this quarter is very close to unemployment last quarter, so the observations are clearly not independent draws, and the series wanders for years above or below its average of 5.88%.
us_macro.csv (BEA and BLS data distributed with statsmodels), as used in Econometrics Made Simple, Chapter 1.Correlation is not causation
Most questions in economics are causal. A finance minister wants to know what will happen to employment if the minimum wage rises. A parent wants to know whether a private school will raise a child’s earnings. Data, however, only show which things move together.
In the wage data, workers with more schooling earn more: the correlation between years of education and the log wage is 0.372. That does not prove that schooling causes higher pay. Three things can create a correlation without a causal effect, or on top of one.
| Source | What happens | Example |
|---|---|---|
| Omitted factors | Something in \( u \) affects \( y \) and is related to \( x \) | More able people stay in school longer and earn more, so a wage–education regression credits schooling with part of the reward for ability (omitted variable bias, Step 6) |
| Reverse causality | \( y \) causes \( x \) | Cities with more police often have more crime, because cities hire police because they have crime |
| Selection | The units that receive \( x \) differ from those that do not | People who choose a training programme may be keener, or more desperate for work, than those who do not |
The ideal experiment
A useful habit is to imagine the experiment you would run if you could. To measure the return to schooling, you would take a large group of children, randomly assign each one a number of years of school, and record their wages twenty years later. Random assignment makes ability, family background and motivation the same, on average, in every schooling group, so any wage difference must be caused by the schooling.
That experiment is impossible and unethical. Econometrics offers ways to get closer to it with observed data: controlling for more variables (multiple regression, Step 6), using natural experiments and instruments (Step 13), comparing units with themselves over time (Step 15), and comparing groups before and after a policy (Step 16).
Across countries, those with more mobile phones per person have higher life expectancy. A newspaper concludes that handing out phones would make people live longer. Evaluate the claim.
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Your first regression
Before any regression, look at the data. In wages.csv, the average hourly wage is $20.48, with a long right tail: the median is $17.61 and the highest wage is $138.54. Average schooling is 11.64 years, ranging from 8 to 18, and 51.4% of the sample are women. Now fit the simple model
Econometrics Made Simple runs this in Python, R and Stata, and all three give the same numbers. In Python it is one line with statsmodels: smf.ols("np.log(wage) ~ educ", data=wages).fit(). The fitted equation is
| Number | Value | What it says |
|---|---|---|
| Slope | 0.1026 | Each extra year of education is associated with a wage about 10.3% higher. |
| Standard error of the slope | 0.0081 | A typical sampling error for the slope (Step 4). |
| \( t \) statistic | 12.68 | The slope is 12.68 standard errors from zero; the \( p \)-value is below 0.001 (Step 5). |
| \( R^2 \) | 0.139 | Education alone accounts for about 14% of the variation in log wages; the other 86% is experience, gender, region, ability, luck and everything else in \( u \). |
wages.csv; bigger dots have more workers behind them. The fitted line rises by 0.103 per year of school. Data: Econometrics Made Simple, Chapter 1.Figure 1 shows what the line is doing. It runs through the average log wage at each level of schooling, which rise fairly steadily from 2.395 at 8 years to 3.250 at 16 years. Above 15 years there are so few workers (6, 5 and 1) that those group averages bounce around; the line borrows strength from all 1,000 workers instead. Step 2 shows that these group averages are estimates of the conditional expectation, the idea that defines a regression.
Use the fitted equation to predict the hourly wage of a worker with 12 years of education and of one with 16 years. By what percentage do the predictions differ?
Four extra years of school are associated with predicted wages 51% higher. Use \( 100\beta_1 \) as a percentage only for small changes; for big ones, use \( 100(e^{\beta_1 \Delta x}-1) \).
The notation you will use
This guide uses the standard notation of econometrics, so that you can move easily between textbooks and research papers.
| Symbol | Meaning |
|---|---|
| \( y_i,\ x_i \) | the values of \( y \) and \( x \) for observation \( i \) |
| \( n \) | the number of observations in the sample |
| \( k \) | the number of explanatory variables, not counting the intercept |
| \( \beta_j,\ \hat\beta_j \) | a population parameter and its estimate |
| \( u_i \) | the error for observation \( i \) (never observed) |
| \( \hat y_i,\ \hat u_i \) | the fitted value and the residual, \( \hat u_i = y_i – \hat y_i \) |
| \( \bar y \) | the sample mean of \( y \) |
| \( t \) | the time period, used instead of \( i \) in time series |
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Practise: 6 questions
Try each one on paper first, then check the worked answer.
EasyQ1. Classify each dataset as a cross-section, time series, pooled cross-section or panel. (a) Monthly UK inflation from 2000 to 2024. (b) A survey of 3,000 households carried out once in 2023. (c) Exam results for the same 200 schools every year from 2015 to 2024. (d) Separate random samples of firms in 2012 and 2022. Show worked answer
(b) Cross-section: many households, one period.
(c) Panel: the same schools are followed over time.
(d) Pooled cross-section: the firms are different in the two years.
EasyQ2. A researcher models \( \text{score} = \beta_0 + \beta_1\,\text{attendance} + u \), where score is a student’s exam mark and attendance is the share of lectures attended. (a) What sign do you expect for \( \beta_1 \)? (b) Name two factors in \( u \). (c) Is either likely to be related to attendance? Show worked answer
(b) For example motivation, prior ability, hours of paid work, distance from campus.
(c) Yes. Motivated students attend more and study harder outside lectures, so motivation is in \( u \) and positively related to attendance. Paid work is likely to lower both. Either way, \( \hat\beta_1 \) mixes the effect of lectures with the effect of the omitted factors.
MediumQ3. Using \( \widehat{\log(\text{wage})} = 1.690 + 0.103\,\text{educ} \), predict the hourly wage of workers with 9 and 11 years of education, and find the exact percentage difference between them. Show worked answer
11 years: \( 1.690 + 0.103 \times 11 = 2.823 \), so \( e^{2.823} = \$16.83 \).
Log difference \( = 0.103 \times 2 = 0.206 \).
Exact difference \( = 100(e^{0.206}-1) = 22.9\% \). (Check: \( 16.83/13.69 = 1.229 \).) The rough reading, 20.6%, understates it.
MediumQ4. Firms that spend more on staff training have higher output per worker. A consultant concludes that every firm should double its training budget. Give one reason based on reverse causality and one based on an omitted factor why the conclusion may not follow. Show worked answer
Omitted factor: good management (in \( u \)) raises productivity directly and also makes a firm more likely to train its staff. The correlation then credits training with part of the effect of management.
Either way, the raw correlation overstates what doubling the budget would achieve.
MediumQ5. In \( \log(\text{earnings}) = \beta_0 + \beta_1\,\text{exper} + u \), the estimate is \( \hat\beta_1 = 0.025 \). (a) Interpret it. (b) Using the rough rule and the exact formula, by what percentage are predicted earnings higher for a worker with 12 more years of experience? Show worked answer
(b) Rough rule: \( 2.5 \times 12 = 30\% \).
Exact: \( 100(e^{0.025 \times 12}-1) = 100(e^{0.3}-1) = 35.0\% \).
For a change this large, use the exact figure.
ExamQ6. A researcher has data on the employment rate of 16 to 19-year-olds and the minimum wage in 50 states for each year from 2010 to 2019. (a) What type of data is this? [1] (b) Write an econometric model for the effect of the minimum wage on teenage employment, and interpret its slope. [2] (c) Describe the ideal experiment. [2] (d) Explain why a state’s minimum wage may be related to the error term. [3] Show worked answer
(b) \( \text{emp} = \beta_0 + \beta_1\,\text{minwage} + u \) [1]; \( \beta_1 \) is the change in the teenage employment rate (percentage points) for a $1 higher minimum wage, holding the factors in \( u \) fixed. [1]
(c) Randomly assign different minimum wages to otherwise similar states [1] and compare later teenage employment; randomisation makes the other factors the same on average. [1]
(d) States tend to raise the minimum wage when their labour markets are strong [1], so the state of the local economy is in \( u \) and positively related to minwage [1]. A strong economy also raises teenage employment, so \( \hat\beta_1 \) is biased upwards and could hide a negative effect. [1]
Total: 8 marks.
Check yourself
1. GDP growth for the same 60 countries every year from 2000 to 2019 is
2. In \( y = \beta_0 + \beta_1 x + u \), the error term \( u \) contains
3. In \( \log(\text{wage}) = \beta_0 + \beta_1\,\text{educ} + u \), \( \hat\beta_1 = 0.08 \) means one more year of education is associated with
Your score: 0 out of 3
The short version
- Econometrics measures economic relationships, tests theories and forecasts, mostly with non-experimental data. Regression is the central tool.
- An empirical study goes: question, economic model, econometric model, data, estimation, testing, interpretation. Failed checks send you back to the model.
- An econometric model has parameters, variables and an error term \( u \). Whether anything in \( u \) is related to the \( x \) variables decides whether estimates can be read as causal.
- Data come as cross-sections, time series, pooled cross-sections and panels; count the units and the periods per unit.
- Omitted factors, reverse causality and selection all create correlations without causation. The randomised experiment is the benchmark.
- In the companion wage data, one more year of school is associated with a wage about 10.3% higher, and education alone explains about 14% of the variation in log wages.
- Chapter 1 of Econometrics Made Simple has the full Python, R and Stata code, more worked examples and 17 exercises with answers.
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