Simultaneous Equations & Time Series

Part of: Econometrics Lecture 06 — Applied Econometrics, Dr. Aluma Dembo Key concepts: Simultaneous Equations Model, Serial Correlation, Time Series, Distributed Lag Model, Autoregressive Model, Static Model, Strict Exogeneity, Seasonality, HAC Standard Errors

Part 1: Simultaneous Equations Models

The Fulton Fish Market Problem

Graddy (1995) studied the wholesale market for whiting fish at the Fulton Fish Market in New York (Dec 1991 – May 1992). The goal: estimate supply and demand elasticities — how sensitive quantity is to price on each side of the market.

The model has two equations:

• $\alpha_S$ = price elasticity of supply (how much more fish gets brought if price rises)
• $\alpha_D$ = price elasticity of demand (how much less fish gets bought if price rises — expected negative)
• At equilibrium: $Q_t^S = Q_t^D = Q_t$

The supply/demand logic

Think of the supply and demand curves as fixed relationships between price and quantity. What we observe in the data is the equilibrium point where those curves cross. Over time, these curves shift due to weather (supply shifts) and day of the week (demand shifts) — and those shifts trace out different equilibrium points.

Why OLS Fails: Endogeneity in Simultaneous Systems

The central problem: price $P_t$ is endogenous in both equations.

Here's why: if there's a random shock to supply ($u_t^S$ changes), the supply curve shifts, the equilibrium moves, and so price changes. This means $\text{cov}(P_t, u_t^S) \neq 0$. Similarly, a demand shock shifts the demand curve and changes equilibrium price, so $\text{cov}(P_t, u_t^D) \neq 0$.

In other words, price and quantity are determined together — they're jointly endogenous. Running OLS on either equation treats price as exogenous, which it isn't.

The identification problem

When you observe price and quantity over time, you're watching equilibrium points shift around. Those shifts could be driven by supply changes or demand changes — and without extra information, you can't tell which curve you're tracing. This is the classic "identification problem" in supply/demand estimation.

  Supply shifts (weather) → supply curve moves → new equilibrium (P,Q)
  Demand shifts (day of week) → demand curve moves → new equilibrium (P,Q)

IV Solution: Using Curve Shifters as Instruments

The fix: use variables that shift one curve as instruments for price. This traces out the other curve.

Identifying the Demand Curve

To estimate the demand elasticity, we need exogenous variation in price that's not caused by demand. Weather (affecting supply only) gives us exactly this.

First stage — instrument for equilibrium quantity using weather:

Second stage — demand equation using predicted quantity:

Why does this work?

• Weather affects supply (rougher seas = fewer fish) → shifts the supply curve → price changes for supply-side reasons
• Weather has no direct effect on demand (consumers don't care about sea conditions)
• So weather is a valid instrument for price in the demand equation

The logic in plain English

On stormy days, fewer fish make it to market. Supply drops, price rises. This price variation is purely supply-driven — it's not because consumers suddenly want more fish. So by looking at how quantity demanded responds to these supply-driven price changes, we can estimate the demand elasticity. That's what IV achieves.

type: supply-shift-identification

Why a shifter "traces out" the other curve

Hold demand fixed (green) and let weather push the supply curve (blue) in and out. Each storm or calm day produces a new equilibrium where the curves cross — and those red dots line up exactly along the demand curve. That's the whole trick: weather varies price for supply-side reasons only, so watching how quantity responds to those price moves reveals the demand elasticity. Without a shifter you'd just see a scatter of dots and couldn't tell which curve you were looking at.

Part 2: Serial Correlation in Time Series

What is Serial Correlation?

Serial correlation (also called autocorrelation) is when residuals at different time periods are correlated with each other:

In the fish market example: after running the IV model, plotting residuals over time reveals a pattern — today's residual is related to yesterday's residual. This violates Assumption TS.4 (no serial correlation).

type: serial-correlation

Two views of the same problem

On the left, the residuals don't bounce randomly around zero — they drift in long runs above and below the line (a positive shock today tends to persist tomorrow). On the right, regressing $\hat u_t$ on $\hat u_{t-1}$ gives a clear positive slope ($\hat\rho$), so we reject $H_0:\rho=0$. The coefficients stay consistent, but the naive standard errors are too small — which is exactly what HAC (Newey-West) errors repair.

Why would errors be serially correlated?

In time series, many things that affect your outcome today also affect it tomorrow. If today's fish price is unexpectedly high due to some unobserved factor (e.g. a food festival this week), that effect often carries over into tomorrow's price. The error term "remembers" the past.

Why is this a problem?

Serial correlation invalidates your t-tests

Even if your coefficients are right in large samples, the standard errors you compute assuming no serial correlation are biased downward — making your estimates look more significant than they are. This is why you can't just ignore it.

Fix: HAC Standard Errors (Newey-West)

Use Heteroskedasticity and Autocorrelation Consistent (HAC) standard errors. The most common are Newey-West standard errors.

• The coefficient estimates don't change — only the standard errors are recalculated
• HAC standard errors are typically larger than naive OLS standard errors (correct widening of confidence intervals)
• p-values increase → some previously significant results may no longer be significant

Testing for Serial Correlation

Run a regression of residuals on their own lag:

Test $H_0: \rho = 0$ against $H_1: \rho \neq 0$.

• If you reject $H_0$: evidence of serial correlation → use HAC standard errors.

Part 3: Time Series Models

What is a Time Series?

A time series is a sequence of random variables indexed by time: $\{y_t\}_{t=1}^T$.

• What you observe is one realisation (one possible sequence of outcomes) from an underlying stochastic process
• The "population" is all possible sequences the process could have generated
• The key feature: ordering matters — the past can influence the future, but not vice versa

Why is time series different from cross-section?

With cross-sectional data (e.g. 1,000 people), observations are typically independent. With time series, observations are correlated over time — what happened last quarter shapes what happens this quarter. This requires adapting our assumptions.

The Static Model

The static model assumes $x$ affects $y$ immediately and only in the same time period. No lag effects — whatever $x_t$ is now is what determines $y_t$ now.

Example — Static Phillips Curve:

The idea: higher unemployment today → lower inflation today (firms compete harder for workers, wage pressure drops, prices follow).

Assumptions for OLS with Time Series

In time series, the standard OLS assumptions are modified. The key change is to exogeneity.

TS.2: Strict Exogeneity (stronger than before)

where $\mathbf{X}$ is all explanatory variables across all time periods (past, present, and future).

Compare with the standard cross-section assumption (contemporaneous exogeneity):

Strict exogeneity is a strong demand

Strict exogeneity requires that $u_t$ is uncorrelated with $x_s$ for all $s$ — even future values of $x$. This fails when:

• $x$ is a lagged dependent variable (e.g. $y_{t-1}$ — obviously correlated with $u_{t-1}$ which affects $y_{t-1}$)
• $x$ reacts to past values of $y$ (feedback loops — e.g. central banks adjust interest rates in response to past inflation)

Additional time series assumptions:

• TS.3 Homoskedasticity: $\text{Var}(u_t \mid \mathbf{X}) = \sigma^2$ (constant variance across time)
• TS.4 No serial correlation: $\text{Corr}(u_t, u_s \mid \mathbf{X}) = 0$ for all $s \neq t$
• TS.5 Normality: $u_t \sim \text{i.i.d.} \; N(0, \sigma^2)$

Under A1 + TS.2–TS.4: OLS gives correct standard errors and valid $t$/$F$-tests. Under A1 + TS.2–TS.5: OLS is normally distributed; $t$ and $F$ statistics have exact distributions.

Distributed Lag (DL) Models

What if $x$ in period $t$ affects $y$ not just now, but also in future periods?

This is a Distributed Lag model with $q$ lags. The static model is the special case where $\delta_1 = \cdots = \delta_q = 0$.

Interpreting the coefficients:

• $\delta_0$ = the impact effect — how much $y$ changes in period $t$ if $x$ increases by 1 in period $t$
• $\delta_k$ = the $k$-period lag effect — how much $y$ changes in period $t+k$ due to a one-time increase in $x$ at period $t$
• $\sum_{k=0}^{q} \delta_k$ = the long-run multiplier — the total cumulative effect of a permanent one-unit increase in $x$

Dynamic Phillips Curve

$\text{inflation}_t = \alpha + \delta_0 \text{unemp}_t + \delta_1 \text{unemp}_{t-1} + u_t$

If unemployment increases by 1pp this period:

• Inflation falls by $\delta_0$ this period
• Inflation falls by an additional $\delta_1$ next period
• Total effect over two periods: $\delta_0 + \delta_1$

In the fish market example: $\text{cor}(\text{unemp}_t, \text{unemp}_{t-1}) = 0.752$ — unemployment is highly autocorrelated (persistent), so the static model understates the full effect.

Multicollinearity in DL models

Lagged values of the same variable tend to be highly correlated with each other. This inflates standard errors and makes it hard to identify individual $\delta_k$ coefficients precisely. The long-run total may be well-identified even when individual lags are not.

Autoregressive (AR) Models

What if past values of $y$ itself help predict current $y$?

AR(1) model:

AR(q) model:

Why AR models matter

Many economic series are persistent — GDP this quarter is very similar to GDP last quarter, exchange rates tomorrow look like exchange rates today. AR models capture this persistence directly. They're the backbone of most macroeconomic forecasting.

A critical limitation: strict exogeneity cannot hold

In an AR(1): $y_{t-1}$ is the regressor, and it's correlated with $e_{t-1}$ (because $y_{t-1}$ is partly determined by $e_{t-1}$). So $\text{cov}(y_{t-1}, e_{t-1}) \neq 0$ — strict exogeneity is violated by construction.

This means:

• ✅ AR models are consistent (under weak exogeneity / contemporaneous exogeneity, which does hold)
• ❌ AR models are necessarily biased in finite samples

Weak vs. strict exogeneity in AR models

Weak exogeneity ($\mathbb{E}[e_t \mid y_{t-1}] = 0$) can plausibly hold — knowing last period's $y$ shouldn't help predict this period's error. That's enough for consistency. Strict exogeneity would require $e_t$ to be uncorrelated with future $y$ — impossible if $y_{t+1}$ depends on $y_t$.

Combining AR and DL: The ADL Model

The most general and flexible time series model:

This is an Autoregressive Distributed Lag (ADL) model. It allows for:

• Inertia in $y$ (AR terms)
• Lagged effects of $x$ (DL terms)
• Other controls $Z$

Predicting Global Temperature from CO2

$\text{temp}_t = \alpha + \phi_1 \text{temp}_{t-1} + \phi_2 \text{temp}_{t-2} + \delta_0 \text{CO2}_t + \delta_1 \text{CO2}_{t-1} + \cdots + e_t$ Temperature is persistent (AR terms capture momentum) but also responds to CO2 with a distributed lag effect. ADL handles both.

Seasonality

Many time series exhibit seasonality — predictable, regular patterns tied to the calendar.

Examples:

• Construction activity peaks in summer (weather-dependent)
• Cheese sales spike around Shavuot
• "Matzo" Google searches spike every Passover

Handling seasonality: seasonal dummy variables

For monthly data (January as the reference/control group):

For quarterly data (Q4 as reference):

• Each seasonal dummy captures how much higher/lower the outcome is in that month/quarter compared to the reference period, holding everything else constant.
• The reference category (excluded dummy) is absorbed into $\beta_0$.

Seasonally adjusted data

Some macroeconomic data from statistical agencies (e.g. GDP, unemployment) are already "seasonally adjusted" — the seasonal patterns have been removed before you receive the data. Check your data source: if it's already seasonally adjusted, don't add seasonal dummies again.

Summary: Time Series Model Types

Summary

1. Simultaneous equations models create endogeneity because price and quantity are jointly determined. OLS on either equation is biased.
2. IV solves identification: use supply shifters (weather) to identify demand, and demand shifters (day of week) to identify supply.
3. Serial correlation means residuals are correlated over time — this leaves coefficients consistent but makes standard errors wrong. Fix with HAC (Newey-West) standard errors.
4. Strict exogeneity (TS.2) requires the error to be uncorrelated with $x$ at all time periods — past, present, and future. Contemporaneous exogeneity (A2) only requires independence at the same period.
5. Distributed lag models let $x$ affect $y$ over multiple periods. Each $\delta_k$ is the effect of $x$ at time $t$ on $y$ at time $t+k$.
6. AR models include lagged $y$ as a regressor — consistent but necessarily biased in finite samples.
7. Seasonality is handled with dummy variables for each month or quarter (minus one for the reference category).

Related Notes

• Previous: Lec_05-Sample Selection & Heckman Correction — selection bias, Heckman two-step
• Hub: Econometrics
• Key IV concepts: Instrumental Variables, Endogeneity, Two Stage Least Squares
• Key time series concepts: Serial Correlation, Distributed Lag Model, Autoregressive Model, Strict Exogeneity, HAC Standard Errors, Seasonality
• Methods: Static Model, Simultaneous Equations Model