Chapter 11: Dynamic Models and Distributed Lags

Modelling time dynamics: FDL, ARDL, and long-run effects

Learning Objectives

By the end of this chapter you will be able to:

Distinguish between static and dynamic regression models
Specify and interpret Finite Distributed Lag (FDL) models
Compute impact, interim, and long-run multipliers
Specify and estimate Autoregressive Distributed Lag (ARDL) models using dynlm
Explain why including lagged dependent variables affects serial correlation tests
Interpret short-run and long-run effects in an ARDL model

1 Stationarity of AR Processes

Before specifying dynamic models, we need the concept of stationarity for AR processes, which determines whether OLS on time-series data has standard asymptotic properties.

1.1 Stationary AR(1)

An AR(1) process $y_t = \rho y_{t-1} + e_t$ with $e_t \sim \text{i.i.d.}(0, \sigma_e^2)$ is covariance-stationary if and only if $|\rho| < 1$.

Mean. In stationarity, $E[y_t] = E[y_{t-1}]$. Taking expectations: $\mu = \rho\mu$ implies $\mu(1-\rho) = 0$, so $\mu = 0$ (or incorporate a constant: if $y_t = \alpha + \rho y_{t-1} + e_t$ then $\mu = \alpha/(1-\rho)$).

Variance. $\text{Var}(y_t) = \rho^2 \text{Var}(y_{t-1}) + \sigma_e^2$. In stationarity, $\text{Var}(y_t) = \text{Var}(y_{t-1}) = \gamma_0$:

\[\gamma_0 = \rho^2 \gamma_0 + \sigma_e^2 \implies \gamma_0 = \frac{\sigma_e^2}{1 - \rho^2}\]

This is finite if and only if $|\rho| < 1$.

Autocovariance. $\gamma(h) = \text{Cov}(y_t, y_{t-h}) = \rho^h \gamma_0$, so the autocorrelation function is:

\[\text{Corr}(y_t, y_{t-h}) = \rho^h\]

The ACF decays geometrically — slowly for $\rho$ near 1, quickly for $\rho$ near 0. For $\rho = 1$ (a random walk), $\gamma_0 = \infty$ and the ACF does not decay (the process is non-stationary).

1.2 AR(p) and the Stationarity Condition

A general AR($p$) process: $y_t = \rho_1 y_{t-1} + \cdots + \rho_p y_{t-p} + e_t$ is stationary if and only if all roots of the characteristic polynomial $1 - \rho_1 z - \cdots - \rho_p z^p = 0$ lie outside the unit circle in the complex plane. For AR(1), this reduces to $|\rho| < 1$.

2 Static vs. Dynamic Models

All the regression models in Chapters 3–10 were static: the effect of $x_t$ on $y_t$ was assumed to occur instantly and completely within period $t$.

In time-series economics, this is often unrealistic. Investment decisions, price adjustments, and behavioural responses all take time. We need dynamic models that capture these lags.

Model type	Specification	Key feature
Static	$y_t = \beta_0 + \beta_1 x_t + u_t$	Instantaneous adjustment
FDL($q$)	$y_t = \alpha_0 + \delta_0 x_t + \delta_1 x_{t-1} + \cdots + \delta_q x_{t-q} + u_t$	Lagged effects of $x$
ARDL($p,q$)	$y_t = \alpha_0 + \sum_{j=1}^p \rho_j y_{t-j} + \sum_{j=0}^q \delta_j x_{t-j} + u_t$	Lagged $y$ and $x$

3 Finite Distributed Lag (FDL) Models

A Finite Distributed Lag model of order $q$ allows $x$ to affect $y$ over $q + 1$ periods:

\[ y_t = \alpha_0 + \delta_0 x_t + \delta_1 x_{t-1} + \delta_2 x_{t-2} + \cdots + \delta_q x_{t-q} + u_t \]

3.1 Multipliers

Multiplier	Definition	Interpretation
Impact multiplier	$\delta_0$	Immediate effect of $\Delta x_t$ on $y_t$
$s$-period multiplier	$\delta_s$	Effect $s$ periods after the change
Interim multiplier (period $m$)	$\sum_{s=0}^m \delta_s$	Cumulative effect after $m+1$ periods
Long-run multiplier (LRM)	$\sum_{s=0}^q \delta_s$	Total cumulative effect of a permanent $\Delta x$

3.2 Example: Fertility and Tax Exemptions

How does a personal tax exemption for children affect the general fertility rate? Economic theory predicts a positive long-run effect (lower cost of children), but the response may be delayed.

# FDL(2) model: fertility on tax exemption with 2 lags
fit_fdl2 <- dynlm(gfr ~ pe + L(pe, 1) + L(pe, 2) + ww2 + pill,
                  data = ts(fertil3))

tidy(fit_fdl2)

b <- coef(fit_fdl2)

impact    <- b["pe"]
interimL1 <- b["pe"] + b["L(pe, 1)"]
lrm       <- b["pe"] + b["L(pe, 1)"] + b["L(pe, 2)"]

cat("Impact multiplier (period 0):", round(impact, 3), "\n")

Impact multiplier (period 0): 0.073

cat("Interim multiplier (period 1):", round(interimL1, 3), "\n")

Interim multiplier (period 1): 0.067

cat("Long-run multiplier (LRM):    ", round(lrm, 3), "\n")

Long-run multiplier (LRM):     0.101

tibble(
  period = 0:2,
  delta  = c(b["pe"], b["L(pe, 1)"], b["L(pe, 2)"]),
  cumulative = cumsum(c(b["pe"], b["L(pe, 1)"], b["L(pe, 2)"]))
) |>
  ggplot(aes(period, delta)) +
  geom_col(fill = "#2c7be5", width = 0.5) +
  geom_line(aes(y = cumulative), colour = "#e63946", linewidth = 1.2) +
  geom_point(aes(y = cumulative), colour = "#e63946", size = 3) +
  scale_x_continuous(breaks = 0:2) +
  labs(x = "Lag (years)", y = "Effect on GFR",
       title = "Distributed Lag Profile: Tax Exemption → Fertility",
       subtitle = "Bars = lag coefficients; red line = cumulative (interim multiplier)")

Figure 1: Distributed lag profile: effect of a unit increase in pe on gfr over time.

Using dynlm

The dynlm package lets you write lag operators directly in the formula: - L(x) = $x_{t-1}$ (one lag) - L(x, 2) = $x_{t-2}$ (two lags) - d(x) = $\Delta x_t = x_t - x_{t-1}$ (first difference) - L(d(x)) = $\Delta x_{t-1}$

This avoids manually creating lag columns and handles the time-series structure automatically.

4 Autoregressive Distributed Lag (ARDL) Models

An ARDL($p, q$) model includes lags of both $y$ and $x$:

\[ y_t = \alpha_0 + \rho_1 y_{t-1} + \cdots + \rho_p y_{t-p} + \delta_0 x_t + \cdots + \delta_q x_{t-q} + u_t \]

The lagged dependent variable $y_{t-1}$ captures persistence — the tendency for $y$ to remain near its past level.

4.1 Why Include Lagged $y$?

Economic theory: partial adjustment, habit formation, adaptive expectations
Serial correlation fix: including $y_{t-1}$ often eliminates AR(1) errors if the static model was misspecified (check with BG test — not DW, which is invalid here)
Parsimonious dynamics: $y_{t-1}$ absorbs the effect of many distant lags of $x$

4.2 Long-Run Multiplier in ARDL

For an ARDL(1,1) model: \[ y_t = \alpha_0 + \rho y_{t-1} + \delta_0 x_t + \delta_1 x_{t-1} + u_t \]

In the long run, $y_t = y_{t-1} = y^*$ and $x_t = x_{t-1} = x^*$. Solving:

\[ \text{LRM} = \frac{\delta_0 + \delta_1}{1 - \rho} \]

The denominator $1 - \rho$ amplifies the long-run effect — the more persistent the process, the larger the long-run multiplier.

4.3 Example: Housing Investment and Interest Rates

hseinv_ts <- ts(hseinv, start = 1947)

# Static model: log investment per capita on log price + time trend
fit_static <- dynlm(log(invpc) ~ log(price) + t, data = hseinv_ts)

# ARDL(1,1): add lagged dependent variable and one lag of log(price)
fit_ardl11 <- dynlm(log(invpc) ~ L(log(invpc)) + log(price) + L(log(price)) + t,
                    data = hseinv_ts)

modelsummary(
  list("Static" = fit_static, "ARDL(1,1)" = fit_ardl11),
  stars   = TRUE,
  gof_map = c("nobs", "r.squared", "adj.r.squared"),
  title   = "Housing Investment: Static vs. ARDL(1,1)"
)

Housing Investment: Static vs. ARDL(1,1)
	Static	ARDL(1,1)
+ p < 0.1, * p < 0.05, p < 0.01, * p < 0.001
(Intercept)	-0.913***	-0.795***
	(0.136)	(0.151)
log(price)	-0.381	2.452*
	(0.679)	(0.933)
t	0.010**	0.011***
	(0.004)	(0.003)
L(log(invpc))		0.344**
		(0.126)
L(log(price))		-3.701***
		(0.929)
Num.Obs.	42	41
R2	0.341	0.639
R2 Adj.	0.307	0.599

b_ardl <- coef(fit_ardl11)
rho_hat <- b_ardl["L(log(invpc))"]
delta0  <- b_ardl["log(price)"]
delta1  <- b_ardl["L(log(price))"]

lrm_ardl <- (delta0 + delta1) / (1 - rho_hat)
cat("Short-run effect of log(price):", round(delta0, 4), "\n")

Short-run effect of log(price): 2.452

cat("Estimated rho (persistence):   ", round(rho_hat, 4), "\n")

Estimated rho (persistence):    0.3445

cat("Long-run multiplier (LRM):     ", round(lrm_ardl, 4), "\n")

Long-run multiplier (LRM):      -1.907

# BG test for serial correlation (DW is invalid with lagged y)
bgtest(fit_ardl11, order = 2)


    Breusch-Godfrey test for serial correlation of order up to 2

data:  fit_ardl11
LM test = 8.7, df = 2, p-value = 0.01

The BG test checks whether adding $y_{t-1}$ has resolved the serial correlation problem. If not significant, the ARDL specification adequately captures the dynamics.

4.4 AR-Error Models vs. ARDL: A Key Distinction

In Chapter 10, we modelled serial correlation in the error term as AR(1): $u_t = \rho u_{t-1} + e_t$. This is different from including a lagged dependent variable.

When the true DGP is ARDL(1,0): $y_t = \alpha + \phi y_{t-1} + \delta x_t + e_t$, but we estimate the static model $y_t = \alpha + \delta x_t + u_t$, the error satisfies $u_t = \phi u_{t-1} + e_t - \phi\delta(x_t - x_{t-1}) + \ldots$. The AR-error structure imposes a geometric lag structure on the FDL: the coefficients on $x_{t-1}, x_{t-2}, \ldots$ must satisfy $\delta_s = \phi^s \delta_0$ — a very restrictive pattern. The ARDL is more flexible: it allows the distributed lag profile to take any shape.

In practice: - If serial correlation arises from model misspecification (missing dynamics), use an ARDL — it fixes the problem and restores valid inference. - If serial correlation arises from data persistence in the error (e.g., measurement error is autocorrelated), then HAC SEs or FGLS are appropriate.

4.5 Central Bank Reaction Function: An ARDL Example

Central banks set interest rates responding to inflation and the output gap, but with inertia — rates do not jump immediately to their target. The standard Taylor rule with interest rate smoothing:

\[i_t = \alpha(1-\rho) + \rho i_{t-1} + \beta_\pi(1-\rho)\pi_t + \beta_y(1-\rho)g_t + e_t\]

where $\rho$ captures smoothing (inertia). This is an ARDL(1,2).

# Use intdef as a proxy: i3 ~ lagged i3 + inflation
data("intdef", package = "wooldridge")
intdef_ts <- ts(intdef, start = 1948)

fit_taylor <- dynlm(i3 ~ L(i3) + inf + def, data = intdef_ts)
tidy(fit_taylor)

b_tr <- coef(fit_taylor)
rho_tr <- b_tr["L(i3)"]
cat("Smoothing parameter rho:       ", round(rho_tr, 4), "\n")

Smoothing parameter rho:        0.7184

cat("Long-run effect of inflation:  ", round(b_tr["inf"] / (1 - rho_tr), 4), "\n")

Long-run effect of inflation:   1.024

cat("Long-run effect of deficit:    ", round(b_tr["def"] / (1 - rho_tr), 4), "\n")

Long-run effect of deficit:     -0.2055

The long-run multiplier on inflation is $\hat{\beta}_{inf}/(1-\hat{\rho})$ — amplified by the persistence of interest rates.

4.6 Partial Adjustment Interpretation

The ARDL(1,0) model has a natural interpretation as a partial adjustment model:

\[ y_t - y_{t-1} = \lambda(y_t^* - y_{t-1}) \quad \text{where} \quad y_t^* = \alpha + \beta x_t \]

If $y$ adjusts only partially toward its equilibrium $y_t^*$ each period, then:

\[ y_t = \lambda \alpha + (1 - \lambda) y_{t-1} + \lambda \beta x_t \]

This is an ARDL(1,0) with $\rho = 1 - \lambda$ (speed of adjustment) and $\delta_0 = \lambda \beta$ (short-run response). The long-run effect is $\beta = \delta_0 / (1 - \rho)$.

5 Static vs. Dynamic: A Comparison

# Simulate impulse responses
shock <- c(1, rep(0, 9))  # one-time shock at t=1

# FDL(3): direct lag effects
fdl_coeffs <- c(0.4, 0.3, 0.2, 0.1)  # illustrative
fdl_response <- convolve(shock, rev(fdl_coeffs), type = "open")[1:10]

# ARDL(1,0): geometric decay
rho_ex  <- 0.7
delta_ex <- 0.5
ardl_response <- numeric(10)
ardl_response[1] <- delta_ex
for (t in 2:10) ardl_response[t] <- rho_ex * ardl_response[t-1]

tibble(
  t       = 1:10,
  FDL     = fdl_response,
  ARDL    = ardl_response
) |>
  pivot_longer(-t, names_to = "model", values_to = "response") |>
  ggplot(aes(t, response, colour = model, group = model)) +
  geom_line(linewidth = 1.2) +
  geom_point(size = 2.5) +
  geom_hline(yintercept = 0, linetype = "dashed") +
  scale_x_continuous(breaks = 1:10) +
  labs(x = "Periods after shock", y = "Response of y",
       colour = "Model",
       title = "Impulse Response Functions",
       subtitle = "FDL = finite lag structure; ARDL = geometric decay")

Figure 2: Impulse response: effect of a unit shock to x in FDL vs. ARDL models.

6 Tutorials

Tutorial 11.1 Using wooldridge::fertil3, estimate an FDL(2) model of the general fertility rate (gfr) on the personal tax exemption (pe) — no controls. Compute and interpret:

The impact multiplier
The 2-period interim multiplier
The long-run multiplier

Is the long-run effect positive, as economic theory predicts?

Solution

fit_fdl2_simple <- dynlm(gfr ~ pe + L(pe, 1) + L(pe, 2), data = ts(fertil3))
tidy(fit_fdl2_simple)

b2 <- coef(fit_fdl2_simple)
cat("Impact multiplier:        ", round(b2["pe"], 3), "\n")

Impact multiplier:         -0.016

cat("2-period interim mult.:   ", round(sum(b2[c("pe","L(pe, 1)","L(pe, 2)")][1:2]), 3), "\n")

2-period interim mult.:    -0.037

cat("Long-run multiplier:      ", round(sum(b2[c("pe","L(pe, 1)","L(pe, 2)")]), 3), "\n")

Long-run multiplier:       0.017

The long-run multiplier should be positive: a permanent $1 increase in the tax exemption per child eventually raises the fertility rate. The impact multiplier may be smaller (or even negative) as people adjust expectations — the full effect accumulates over two years.

Tutorial 11.2 Add a lagged dependent variable to the Phillips curve model (inf ~ unem, wooldridge::phillips) to create an ARDL(1,0). Compare:

Static OLS vs. ARDL(1,0): how do the estimated effects of unemployment on inflation change?
Does the ARDL specification resolve serial correlation? Apply the BG test (order 1).
Compute the long-run effect of unemployment on inflation from the ARDL model.

Solution

fit_static_ph <- dynlm(inf ~ unem, data = ts(phillips, start = 1948))
fit_ardl_ph   <- dynlm(inf ~ L(inf) + unem, data = ts(phillips, start = 1948))

modelsummary(
  list("Static" = fit_static_ph, "ARDL(1,0)" = fit_ardl_ph),
  stars = TRUE, gof_map = c("nobs", "r.squared"),
  title = "Phillips Curve: Static vs. ARDL(1,0)"
)

Phillips Curve: Static vs. ARDL(1,0)
	Static	ARDL(1,0)
+ p < 0.1, * p < 0.05, p < 0.01, * p < 0.001
(Intercept)	1.054	2.210+
	(1.548)	(1.210)
unem	0.502+	-0.224
	(0.266)	(0.239)
L(inf)		0.733***
		(0.117)
Num.Obs.	56	55
R2	0.062	0.477

# BG test on ARDL
bgtest(fit_ardl_ph, order = 1)


    Breusch-Godfrey test for serial correlation of order up to 1

data:  fit_ardl_ph
LM test = 2.2, df = 1, p-value = 0.1

# Long-run multiplier
b_ph <- coef(fit_ardl_ph)
lrm_ph <- b_ph["unem"] / (1 - b_ph["L(inf)"])
cat("LRM (unemployment -> inflation):", round(lrm_ph, 4), "\n")

LRM (unemployment -> inflation): -0.8391

The static model likely has significant serial correlation (BG test). The ARDL model should reduce or eliminate it. The short-run and long-run effects of unemployment on inflation will differ — the long-run multiplier amplifies the short-run coefficient by $1/(1-\hat{\rho})$.

Tutorial 11.3 Explain the difference between the impact multiplier and the long-run multiplier in an FDL model. Under what conditions is the long-run multiplier much larger than the impact multiplier? Provide an economic example.

Solution

The impact multiplier ($\delta_0$) measures the immediate, within-period response of $y$ to a unit change in $x$. The long-run multiplier ($\sum_{s=0}^q \delta_s$) measures the total cumulative response after all lag effects have worked through.

The LRM is much larger than the impact multiplier when: 1. The lag coefficients $\delta_1, \ldots, \delta_q$ are large and positive — the effects of $x$ accumulate over many periods 2. In ARDL models, when $\rho$ (persistence) is close to 1, the denominator $(1-\rho)$ becomes small, amplifying the LRM

Economic example: Consider the effect of a sustained monetary policy change on inflation. In the short run, monetary policy operates with a lag — the impact multiplier might be near zero. Over 12–24 months, however, prices adjust and the full disinflationary effect materialises. The long-run multiplier captures the total reduction in inflation from a permanent 1 percentage point change in the interest rate, which can be several times larger than the impact effect.

--- title: "Chapter 11: Dynamic Models and Distributed Lags" subtitle: "Modelling time dynamics: FDL, ARDL, and long-run effects" --- ```{r} #| label: setup #| include: false library(tidyverse) library(broom) library(modelsummary) library(lmtest) library(sandwich) library(dynlm) library(wooldridge) theme_set(theme_minimal(base_size = 13)) options(digits = 4, scipen = 999) data("phillips", package = "wooldridge") data("hseinv", package = "wooldridge") data("fertil3", package = "wooldridge") ``` ::: {.callout-note} ## Learning Objectives By the end of this chapter you will be able to: - Distinguish between static and dynamic regression models - Specify and interpret Finite Distributed Lag (FDL) models - Compute impact, interim, and long-run multipliers - Specify and estimate Autoregressive Distributed Lag (ARDL) models using `dynlm` - Explain why including lagged dependent variables affects serial correlation tests - Interpret short-run and long-run effects in an ARDL model ::: --- ## Stationarity of AR Processes Before specifying dynamic models, we need the concept of **stationarity** for AR processes, which determines whether OLS on time-series data has standard asymptotic properties. ### Stationary AR(1) An AR(1) process $y_t = \rho y_{t-1} + e_t$ with $e_t \sim \text{i.i.d.}(0, \sigma_e^2)$ is **covariance-stationary** if and only if $|\rho| < 1$. **Mean.** In stationarity, $E[y_t] = E[y_{t-1}]$. Taking expectations: $\mu = \rho\mu$ implies $\mu(1-\rho) = 0$, so $\mu = 0$ (or incorporate a constant: if $y_t = \alpha + \rho y_{t-1} + e_t$ then $\mu = \alpha/(1-\rho)$). **Variance.** $\text{Var}(y_t) = \rho^2 \text{Var}(y_{t-1}) + \sigma_e^2$. In stationarity, $\text{Var}(y_t) = \text{Var}(y_{t-1}) = \gamma_0$: $$\gamma_0 = \rho^2 \gamma_0 + \sigma_e^2 \implies \gamma_0 = \frac{\sigma_e^2}{1 - \rho^2}$$ This is finite if and only if $|\rho| < 1$. **Autocovariance.** $\gamma(h) = \text{Cov}(y_t, y_{t-h}) = \rho^h \gamma_0$, so the **autocorrelation function** is: $$\text{Corr}(y_t, y_{t-h}) = \rho^h$$ The ACF decays geometrically — slowly for $\rho$ near 1, quickly for $\rho$ near 0. For $\rho = 1$ (a random walk), $\gamma_0 = \infty$ and the ACF does not decay (the process is non-stationary). ### AR(p) and the Stationarity Condition A general AR($p$) process: $y_t = \rho_1 y_{t-1} + \cdots + \rho_p y_{t-p} + e_t$ is stationary if and only if all roots of the characteristic polynomial $1 - \rho_1 z - \cdots - \rho_p z^p = 0$ lie **outside the unit circle** in the complex plane. For AR(1), this reduces to $|\rho| < 1$. --- ## Static vs. Dynamic Models All the regression models in Chapters 3–10 were **static**: the effect of $x_t$ on $y_t$ was assumed to occur instantly and completely within period $t$. In time-series economics, this is often unrealistic. Investment decisions, price adjustments, and behavioural responses all take time. We need **dynamic models** that capture these lags. | Model type | Specification | Key feature | |---|---|---| | **Static** | $y_t = \beta_0 + \beta_1 x_t + u_t$ | Instantaneous adjustment | | **FDL($q$)** | $y_t = \alpha_0 + \delta_0 x_t + \delta_1 x_{t-1} + \cdots + \delta_q x_{t-q} + u_t$ | Lagged effects of $x$ | | **ARDL($p,q$)** | $y_t = \alpha_0 + \sum_{j=1}^p \rho_j y_{t-j} + \sum_{j=0}^q \delta_j x_{t-j} + u_t$ | Lagged $y$ and $x$ | --- ## Finite Distributed Lag (FDL) Models A **Finite Distributed Lag model of order $q$** allows $x$ to affect $y$ over $q + 1$ periods: $$ y_t = \alpha_0 + \delta_0 x_t + \delta_1 x_{t-1} + \delta_2 x_{t-2} + \cdots + \delta_q x_{t-q} + u_t $$ ### Multipliers | Multiplier | Definition | Interpretation | |---|---|---| | **Impact multiplier** | $\delta_0$ | Immediate effect of $\Delta x_t$ on $y_t$ | | **$s$-period multiplier** | $\delta_s$ | Effect $s$ periods after the change | | **Interim multiplier** (period $m$) | $\sum_{s=0}^m \delta_s$ | Cumulative effect after $m+1$ periods | | **Long-run multiplier (LRM)** | $\sum_{s=0}^q \delta_s$ | Total cumulative effect of a permanent $\Delta x$ | ### Example: Fertility and Tax Exemptions How does a personal tax exemption for children affect the general fertility rate? Economic theory predicts a positive long-run effect (lower cost of children), but the response may be delayed. ```{r} #| label: fdl-fertility # FDL(2) model: fertility on tax exemption with 2 lags fit_fdl2 <- dynlm(gfr ~ pe + L(pe, 1) + L(pe, 2) + ww2 + pill, data = ts(fertil3)) tidy(fit_fdl2) ``` ```{r} #| label: fdl-multipliers b <- coef(fit_fdl2) impact <- b["pe"] interimL1 <- b["pe"] + b["L(pe, 1)"] lrm <- b["pe"] + b["L(pe, 1)"] + b["L(pe, 2)"] cat("Impact multiplier (period 0):", round(impact, 3), "\n") cat("Interim multiplier (period 1):", round(interimL1, 3), "\n") cat("Long-run multiplier (LRM): ", round(lrm, 3), "\n") ``` ```{r} #| label: fig-lag-profile #| fig-cap: "Distributed lag profile: effect of a unit increase in pe on gfr over time." tibble( period = 0:2, delta = c(b["pe"], b["L(pe, 1)"], b["L(pe, 2)"]), cumulative = cumsum(c(b["pe"], b["L(pe, 1)"], b["L(pe, 2)"])) ) |> ggplot(aes(period, delta)) + geom_col(fill = "#2c7be5", width = 0.5) + geom_line(aes(y = cumulative), colour = "#e63946", linewidth = 1.2) + geom_point(aes(y = cumulative), colour = "#e63946", size = 3) + scale_x_continuous(breaks = 0:2) + labs(x = "Lag (years)", y = "Effect on GFR", title = "Distributed Lag Profile: Tax Exemption → Fertility", subtitle = "Bars = lag coefficients; red line = cumulative (interim multiplier)") ``` ::: {.callout-tip} ## Using `dynlm` The `dynlm` package lets you write lag operators directly in the formula: - `L(x)` = $x_{t-1}$ (one lag) - `L(x, 2)` = $x_{t-2}$ (two lags) - `d(x)` = $\Delta x_t = x_t - x_{t-1}$ (first difference) - `L(d(x))` = $\Delta x_{t-1}$ This avoids manually creating lag columns and handles the time-series structure automatically. ::: --- ## Autoregressive Distributed Lag (ARDL) Models An **ARDL($p, q$)** model includes lags of both $y$ and $x$: $$ y_t = \alpha_0 + \rho_1 y_{t-1} + \cdots + \rho_p y_{t-p} + \delta_0 x_t + \cdots + \delta_q x_{t-q} + u_t $$ The lagged dependent variable $y_{t-1}$ captures **persistence** — the tendency for $y$ to remain near its past level. ### Why Include Lagged $y$? 1. **Economic theory**: partial adjustment, habit formation, adaptive expectations 2. **Serial correlation fix**: including $y_{t-1}$ often eliminates AR(1) errors if the static model was misspecified (check with BG test — not DW, which is invalid here) 3. **Parsimonious dynamics**: $y_{t-1}$ absorbs the effect of many distant lags of $x$ ### Long-Run Multiplier in ARDL For an ARDL(1,1) model: $$ y_t = \alpha_0 + \rho y_{t-1} + \delta_0 x_t + \delta_1 x_{t-1} + u_t $$ In the long run, $y_t = y_{t-1} = y^*$ and $x_t = x_{t-1} = x^*$. Solving: $$ \text{LRM} = \frac{\delta_0 + \delta_1}{1 - \rho} $$ The denominator $1 - \rho$ amplifies the long-run effect — the more persistent the process, the larger the long-run multiplier. ### Example: Housing Investment and Interest Rates ```{r} #| label: ardl-housing hseinv_ts <- ts(hseinv, start = 1947) # Static model: log investment per capita on log price + time trend fit_static <- dynlm(log(invpc) ~ log(price) + t, data = hseinv_ts) # ARDL(1,1): add lagged dependent variable and one lag of log(price) fit_ardl11 <- dynlm(log(invpc) ~ L(log(invpc)) + log(price) + L(log(price)) + t, data = hseinv_ts) modelsummary( list("Static" = fit_static, "ARDL(1,1)" = fit_ardl11), stars = TRUE, gof_map = c("nobs", "r.squared", "adj.r.squared"), title = "Housing Investment: Static vs. ARDL(1,1)" ) ``` ```{r} #| label: ardl-lrm b_ardl <- coef(fit_ardl11) rho_hat <- b_ardl["L(log(invpc))"] delta0 <- b_ardl["log(price)"] delta1 <- b_ardl["L(log(price))"] lrm_ardl <- (delta0 + delta1) / (1 - rho_hat) cat("Short-run effect of log(price):", round(delta0, 4), "\n") cat("Estimated rho (persistence): ", round(rho_hat, 4), "\n") cat("Long-run multiplier (LRM): ", round(lrm_ardl, 4), "\n") ``` ```{r} #| label: ardl-sc-test # BG test for serial correlation (DW is invalid with lagged y) bgtest(fit_ardl11, order = 2) ``` The BG test checks whether adding $y_{t-1}$ has resolved the serial correlation problem. If not significant, the ARDL specification adequately captures the dynamics. ### AR-Error Models vs. ARDL: A Key Distinction In Chapter 10, we modelled serial correlation in the **error term** as AR(1): $u_t = \rho u_{t-1} + e_t$. This is different from including a lagged dependent variable. When the true DGP is ARDL(1,0): $y_t = \alpha + \phi y_{t-1} + \delta x_t + e_t$, but we estimate the static model $y_t = \alpha + \delta x_t + u_t$, the error satisfies $u_t = \phi u_{t-1} + e_t - \phi\delta(x_t - x_{t-1}) + \ldots$. The AR-error structure imposes a **geometric lag structure** on the FDL: the coefficients on $x_{t-1}, x_{t-2}, \ldots$ must satisfy $\delta_s = \phi^s \delta_0$ — a very restrictive pattern. The ARDL is more flexible: it allows the distributed lag profile to take any shape. In practice: - If serial correlation arises from **model misspecification** (missing dynamics), use an ARDL — it fixes the problem and restores valid inference. - If serial correlation arises from **data persistence** in the error (e.g., measurement error is autocorrelated), then HAC SEs or FGLS are appropriate. ### Central Bank Reaction Function: An ARDL Example Central banks set interest rates responding to inflation and the output gap, but with **inertia** — rates do not jump immediately to their target. The standard Taylor rule with interest rate smoothing: $$i_t = \alpha(1-\rho) + \rho i_{t-1} + \beta_\pi(1-\rho)\pi_t + \beta_y(1-\rho)g_t + e_t$$ where $\rho$ captures smoothing (inertia). This is an ARDL(1,2). ```{r} #| label: taylor-rule # Use intdef as a proxy: i3 ~ lagged i3 + inflation data("intdef", package = "wooldridge") intdef_ts <- ts(intdef, start = 1948) fit_taylor <- dynlm(i3 ~ L(i3) + inf + def, data = intdef_ts) tidy(fit_taylor) b_tr <- coef(fit_taylor) rho_tr <- b_tr["L(i3)"] cat("Smoothing parameter rho: ", round(rho_tr, 4), "\n") cat("Long-run effect of inflation: ", round(b_tr["inf"] / (1 - rho_tr), 4), "\n") cat("Long-run effect of deficit: ", round(b_tr["def"] / (1 - rho_tr), 4), "\n") ``` The long-run multiplier on inflation is $\hat{\beta}_{inf}/(1-\hat{\rho})$ — amplified by the persistence of interest rates. ### Partial Adjustment Interpretation The ARDL(1,0) model has a natural interpretation as a **partial adjustment model**: $$ y_t - y_{t-1} = \lambda(y_t^* - y_{t-1}) \quad \text{where} \quad y_t^* = \alpha + \beta x_t $$ If $y$ adjusts only partially toward its equilibrium $y_t^*$ each period, then: $$ y_t = \lambda \alpha + (1 - \lambda) y_{t-1} + \lambda \beta x_t $$ This is an ARDL(1,0) with $\rho = 1 - \lambda$ (speed of adjustment) and $\delta_0 = \lambda \beta$ (short-run response). The long-run effect is $\beta = \delta_0 / (1 - \rho)$. --- ## Static vs. Dynamic: A Comparison ```{r} #| label: fig-dynamic-comparison #| fig-cap: "Impulse response: effect of a unit shock to x in FDL vs. ARDL models." # Simulate impulse responses shock <- c(1, rep(0, 9)) # one-time shock at t=1 # FDL(3): direct lag effects fdl_coeffs <- c(0.4, 0.3, 0.2, 0.1) # illustrative fdl_response <- convolve(shock, rev(fdl_coeffs), type = "open")[1:10] # ARDL(1,0): geometric decay rho_ex <- 0.7 delta_ex <- 0.5 ardl_response <- numeric(10) ardl_response[1] <- delta_ex for (t in 2:10) ardl_response[t] <- rho_ex * ardl_response[t-1] tibble( t = 1:10, FDL = fdl_response, ARDL = ardl_response ) |> pivot_longer(-t, names_to = "model", values_to = "response") |> ggplot(aes(t, response, colour = model, group = model)) + geom_line(linewidth = 1.2) + geom_point(size = 2.5) + geom_hline(yintercept = 0, linetype = "dashed") + scale_x_continuous(breaks = 1:10) + labs(x = "Periods after shock", y = "Response of y", colour = "Model", title = "Impulse Response Functions", subtitle = "FDL = finite lag structure; ARDL = geometric decay") ``` --- ## Tutorials **Tutorial 11.1** Using `wooldridge::fertil3`, estimate an FDL(2) model of the general fertility rate (`gfr`) on the personal tax exemption (`pe`) — no controls. Compute and interpret: a. The impact multiplier b. The 2-period interim multiplier c. The long-run multiplier Is the long-run effect positive, as economic theory predicts? ::: {.callout-tip collapse="true"} ## Solution ```{r} #| label: ex11-1 fit_fdl2_simple <- dynlm(gfr ~ pe + L(pe, 1) + L(pe, 2), data = ts(fertil3)) tidy(fit_fdl2_simple) b2 <- coef(fit_fdl2_simple) cat("Impact multiplier: ", round(b2["pe"], 3), "\n") cat("2-period interim mult.: ", round(sum(b2[c("pe","L(pe, 1)","L(pe, 2)")][1:2]), 3), "\n") cat("Long-run multiplier: ", round(sum(b2[c("pe","L(pe, 1)","L(pe, 2)")]), 3), "\n") ``` The long-run multiplier should be positive: a permanent $1 increase in the tax exemption per child eventually raises the fertility rate. The impact multiplier may be smaller (or even negative) as people adjust expectations — the full effect accumulates over two years. ::: **Tutorial 11.2** Add a lagged dependent variable to the Phillips curve model (`inf ~ unem`, `wooldridge::phillips`) to create an ARDL(1,0). Compare: a. Static OLS vs. ARDL(1,0): how do the estimated effects of unemployment on inflation change? b. Does the ARDL specification resolve serial correlation? Apply the BG test (order 1). c. Compute the long-run effect of unemployment on inflation from the ARDL model. ::: {.callout-tip collapse="true"} ## Solution ```{r} #| label: ex11-2 fit_static_ph <- dynlm(inf ~ unem, data = ts(phillips, start = 1948)) fit_ardl_ph <- dynlm(inf ~ L(inf) + unem, data = ts(phillips, start = 1948)) modelsummary( list("Static" = fit_static_ph, "ARDL(1,0)" = fit_ardl_ph), stars = TRUE, gof_map = c("nobs", "r.squared"), title = "Phillips Curve: Static vs. ARDL(1,0)" ) # BG test on ARDL bgtest(fit_ardl_ph, order = 1) # Long-run multiplier b_ph <- coef(fit_ardl_ph) lrm_ph <- b_ph["unem"] / (1 - b_ph["L(inf)"]) cat("LRM (unemployment -> inflation):", round(lrm_ph, 4), "\n") ``` The static model likely has significant serial correlation (BG test). The ARDL model should reduce or eliminate it. The short-run and long-run effects of unemployment on inflation will differ — the long-run multiplier amplifies the short-run coefficient by $1/(1-\hat{\rho})$. ::: **Tutorial 11.3** Explain the difference between the impact multiplier and the long-run multiplier in an FDL model. Under what conditions is the long-run multiplier much larger than the impact multiplier? Provide an economic example. ::: {.callout-tip collapse="true"} ## Solution The **impact multiplier** ($\delta_0$) measures the immediate, within-period response of $y$ to a unit change in $x$. The **long-run multiplier** ($\sum_{s=0}^q \delta_s$) measures the total cumulative response after all lag effects have worked through. The LRM is much larger than the impact multiplier when: 1. The lag coefficients $\delta_1, \ldots, \delta_q$ are large and positive — the effects of $x$ accumulate over many periods 2. In ARDL models, when $\rho$ (persistence) is close to 1, the denominator $(1-\rho)$ becomes small, amplifying the LRM **Economic example:** Consider the effect of a sustained monetary policy change on inflation. In the short run, monetary policy operates with a lag — the impact multiplier might be near zero. Over 12–24 months, however, prices adjust and the full disinflationary effect materialises. The long-run multiplier captures the total reduction in inflation from a permanent 1 percentage point change in the interest rate, which can be several times larger than the impact effect. :::

Model type	Specification	Key feature
Static	\(y_t = \beta_0 + \beta_1 x_t + u_t\)	Instantaneous adjustment
FDL(\(q\))	\(y_t = \alpha_0 + \delta_0 x_t + \delta_1 x_{t-1} + \cdots + \delta_q x_{t-q} + u_t\)	Lagged effects of \(x\)
ARDL(\(p,q\))	\(y_t = \alpha_0 + \sum_{j=1}^p \rho_j y_{t-j} + \sum_{j=0}^q \delta_j x_{t-j} + u_t\)	Lagged \(y\) and \(x\)

Multiplier	Definition	Interpretation
Impact multiplier	\(\delta_0\)	Immediate effect of \(\Delta x_t\) on \(y_t\)
\(s\)-period multiplier	\(\delta_s\)	Effect \(s\) periods after the change
Interim multiplier (period \(m\))	\(\sum_{s=0}^m \delta_s\)	Cumulative effect after \(m+1\) periods
Long-run multiplier (LRM)	\(\sum_{s=0}^q \delta_s\)	Total cumulative effect of a permanent \(\Delta x\)

1 Stationarity of AR Processes

1.1 Stationary AR(1)

1.2 AR(p) and the Stationarity Condition

2 Static vs. Dynamic Models

3 Finite Distributed Lag (FDL) Models

3.1 Multipliers

3.2 Example: Fertility and Tax Exemptions

4 Autoregressive Distributed Lag (ARDL) Models

4.1 Why Include Lagged \(y\)?

4.2 Long-Run Multiplier in ARDL

4.3 Example: Housing Investment and Interest Rates

4.4 AR-Error Models vs. ARDL: A Key Distinction

4.5 Central Bank Reaction Function: An ARDL Example

4.6 Partial Adjustment Interpretation

5 Static vs. Dynamic: A Comparison

6 Tutorials