ECON 0150 | Economic Data Analysis

The economist’s data analysis skillset.


Part 4.4 | The Problem of Timeseries

Part 4.3 Recap

We check four assumptions to trust our model’s results.


  1. Linearity: The relationship between X and Y is linear
  2. Homoskedasticity: Equal error variance across all values of X
  3. Independence: Observations are independent from each other
  4. Normality: Errors are normally distributed


Today we focus on Assumption 3: Independence: the problem of timeseries.

Timeseries Analysis

We often want to model relationships through time.

Data related in time has a special problem:

  • Observations are related to their past values (autocorrelation).
  • This violates Assumption 4: Independence.

Timeseries Analysis: Autocorrelation

We can check whether values in timeseries are related to their own past values.

A Lag Plot shows the value today (\(t\)) against the value yesterday (\(t-1\)).

Autocorrelation tells us whether today’s value depends on yesterday’s value.

Timeseries: Model 1 (Levels)

The standard approach has problems with time series.

\[\text{Y} = \beta_0 + \beta_1 \cdot \text{t} + \varepsilon\]

Timeseries: Model 1 (Levels)

The standard approach has problems with time series.

\[\text{Y} = \beta_0 + \beta_1 \cdot \text{t} + \varepsilon\]

This model is often wrong in the same direction repeatedly.

Timeseries: Model 1 (Levels)

This ‘levels’ model shows strong patterns in residuals (autocorrelation).

\[\text{Y} = \beta_0 + \beta_1 \cdot \text{t} + \varepsilon\]

Exercise 4.4 | Model 1 (Levels)

Okun’s Law: when unemployment rises, GDP tends to fall. 

# 1. Fit the levels model
model1 = smf.ols('gdp ~ unemployment', data=data).fit()
print(model1.summary().tables[1])


# 2. Residual plot
plt.scatter(model1.predict(), model1.resid)
plt.axhline(0, color='red')


# 3. Lagged residual plot
plt.scatter(model1.resid[:-1], model1.resid[1:])

Timeseries: Model 2 (First Differences)

We can fix some issues of autocorrelation by looking at changes instead of levels.

\[\Delta \text{Y}_t = \text{Y}_t - \text{Y}_{t-1}\]

Looking at changes instead of levels shows no relationship (correctly here).

What would first differences look like if there WAS a positive trend?

Timeseries: Model 2 (First Differences)

What would first differences look like if there WAS a positive trend?

Timeseries: Model 2 (First Differences)

What would first differences look like if there WAS a positive trend?

The vertical intercept \(\beta_0\) is positive!

The slope coefficient \(\beta_1\) is zero!

Timeseries: Model 2 (First Differences)

First differences reduces but does not eliminate the problem of autocorrelation.

Exercise 4.4 | Model 2

Examine a first difference model of the relationship between GDP and unemployment. 


# Step 1. Create first differences of Y
data['gdp_diff'] = data['gdp'].diff()
data = data.dropna()


# Step 2. Fit the model (Y differenced, X in levels)
model2 = smf.ols('gdp_diff ~ unemployment', data=data).fit()
print(model2.summary().tables[1])
# Step 3. Residual and lagged residual plots

Timeseries: Model 3 (Double First Differences)

Sometimes we want to measure how two variables move together.

\[\Delta \text{Y}_t = \beta_0 + \beta_1 \times \Delta \text{X}_t + \varepsilon_t\]

Timeseries: Model 3 (Double First Differences)

Relating changes in X to changes in Y.

\[\Delta \text{Y}_t = \beta_0 + \beta_1 \times \Delta \text{X}_t + \varepsilon_t\]


  • Further reduces serial correlation in the error terms.
  • \(\beta_0\) captures time trend in \(Y\)
  • \(\beta_1\) captures the short-term relationship between variables.
  • Clear interpretation: how do changes in X relate to changes in Y?

Exercise 4.4 | Model 3

Examine a double first difference model of the relationship between GDP and unemployment. 


# Step 1. Create first differences of X
data['unemployment_diff'] = data['unemployment'].diff()
data = data.dropna()


# Step 2. Fit the double differences model (both X and Y differenced)
model3 = smf.ols('gdp_diff ~ unemployment_diff', data=data).fit()
print(model3.summary().tables[1])


# Step 3. Residual and lagged residual plots

\(\beta_1\) now represents the short-term relationship between changes in X and Y

Timeseries: Model 4 (Growth Rates)

Proportional changes provide interpretable coefficients:

\[g_Y = \frac{\text{Y}_t - \text{Y}_{t-1}}{\text{Y}_{t-1}} = \frac{\Delta \text{Y}_t}{\text{Y}_{t-1}}\]

Timeseries: Model 4 (Growth Rates)

Proportional changes provide interpretable coefficients:

\[g_Y = \frac{\text{Y}_t - \text{Y}_{t-1}}{\text{Y}_{t-1}} = \frac{\Delta \text{Y}_t}{\text{Y}_{t-1}}\]

Timeseries: Model 4 (Growth Rates)

Proportional changes provide interpretable coefficients:

\[g_Y = \frac{\text{Y}_t - \text{Y}_{t-1}}{\text{Y}_{t-1}} = \frac{\Delta \text{Y}_t}{\text{Y}_{t-1}}\]

Timeseries: Model 4 (Growth Rates)

Is the growth in Y related to the growth in X?

\[g_Y = \beta_0 + \beta_1 \times g_X + \varepsilon_t\]

Timeseries: Model 4 (Growth Rates)

Is the growth in Y related to the growth in X?

\[g_Y = \beta_0 + \beta_1 \times g_X + \varepsilon_t\]


Growth rate models have the advantages of first differences and can scale better.

  • This is natural for variables with exponential growth.
  • \(\beta_0\) is Y’s baseline growth rate.
  • \(\beta_1\) is how Y’s growth responds to a 1 percentage point increase in X’s growth.

Exercise 4.4 | Model 4

Examine a growth rates model of the relationship between GDP and unemployment. 


# Step 1. Calculate growth rates (percentage changes)
data['gdp_growth'] = data['gdp'].pct_change() # in percentage points
data['unemployment_growth'] = data['unemployment'].pct_change()


# Step 2. Drop rows with NaN values
data = data.dropna()


# Step 3. Fit the growth rate model
model4 = smf.ols('gdp_growth ~ unemployment_growth', data=data).fit()
print(model4.summary().tables[1])


# Step 4. Residual and lagged residual plots

\(\beta_1\) is now expressed in percentage point terms

Timeseries: Summary

Autocorrelation can be reduced but not eliminated.


Model Equation \(\beta_1\) Interpretation Best For
Levels \(Y = \beta_0 + \beta_1 X + \varepsilon\) Relationship between levels Cross-sectional data
First Diff \(\Delta Y = \beta_0 + \beta_1 \Delta X + \varepsilon\) Short-term relationship between changes Trending time series
Growth Rates \(g_Y = \beta_0 + \beta_1 g_X + \varepsilon\) Response of \(Y\)’s growth to \(X\)’s growth Exponentially growing series


Always check the residual lag plot. Differencing reduces autocorrelation but rarely eliminates it entirely.