Day 3: Heterogeneity and Dynamics in Data

Dynamic Regression Models

Robert W. Walker

2026-07-22

Outline for Day 3

A Note on Notation from Google AI
The Box-Jenkins Methodology
Dynamic Models
GLS vs. OLS and fixes

A Note on Notation from Google AI

Screenshot

The Box-Jenkins ARIMA Approach

B-J modeling strategy starts with (p, d, q) diagnostics:

e_{t} = \phi_{1} e_{t-1} + \phi_{2}e_{t-2} + \cdots + \phi_{p} e_{t-p} + \nu_{t} + \theta_{1} \nu_{t-1} + \theta_{2}\nu_{t-2} + \cdots + \theta_{q} \nu_{t-q}

Determine Stationarity through a three-pronged strategy:
1. Graph Y_t across time
2. Examine the ACF and PACF
3. Conduct a statistical test for stationarity (e.g., Augmented Dickey-Fuller)
Diagnose the AR and the MA processes
Post-estimation diagnostics of the residuals from a filtering model — if we have the B-J model right, \hat{\epsilon} should be pure random noise.

Portmanteau Tests for White Noise

Goal: White noise residuals – that is, residuals that contain no notable autocorrelation.

Measure: Portmanteau tests are encompassing tests for white noise in a model’s residuals. One of the most popular is the Ljung-Box Q-statistic.
If Y is white noise, then (because correlations are distributed normal, their squares are chi-square):

Q = n(n+2)\sum_{j=1}^{m} \frac{1}{n-j}\hat{\rho}_j^2 \;\rightarrow\; \chi^2_m

where m is the number of autocorrelations calculated (based on the number of lags specified) and \hat{\rho}_j is the estimated autocorrelation for lag j.

Note

Larger Q-statistics indicate that the errors contain some type of systematic process.

Stata’s `corrgram` Output — An Example

LAG    AC       PAC      Q         Prob>Q
  1    0.9458   0.9706   92.164    0.0000
  2    0.8932  -0.0238   175.21    0.0000
  3    0.8523   0.0585   251.59    0.0000
  4    0.8224   0.0526   323.45    0.0000
  5    0.7907   0.0027   390.57    0.0000
  6    0.7457  -0.2053   450.91    0.0000
  7    0.6980  -0.1496   504.35    0.0000
  8    0.6604  -0.0166   552.70    0.0000
  9    0.6229  -0.1212   596.19    0.0000
 10    0.5862  -0.0934   635.14    0.0000
 ...
 40   -0.4325  -0.5224  1035.50    0.0000

Note

Slowly decaying AC values and a single large PAC spike at lag 1 are the hallmark of a non-stationary (unit root) process.

Logging Time Series Variables

Logging variables is a frequently-recommended way to deal with the variance-non-stationarity of a variable.
In practice this doesn’t usually make much of a difference.
Of course if you have done this, you need to keep this in mind when interpreting your results.

Detecting AR and MA Processes

Properties of the ACF and PACF (after Enders Table 2.1)
Model	ACF	PACF
White noise	All zero	All zero
MA(1): \theta > 0	Single positive spike at lag 1	Oscillating decay
MA(1): \theta < 0	Single negative spike at lag 1	Geometric decay
AR(1): \phi_1 > 0	Direct geometric decay	Single spike at lag 1
AR(1): \phi_1 < 0	Oscillating decay	Single spike at lag 1
AR(p)	Decays toward zero	Spikes through lag p; zero for s > p
ARMA(1,1)	Geometric decay after lag 1	Oscillating decay after lag 1
ARMA(p,q)	Decay after lag q	Decay after lag p

Detecting AR and MA Processes — Rules of Thumb

Pure AR processes generally lead to declining ACFs and spikes in the PACF to indicate the ordering.
Pure MA processes show the opposite — a spike in the ACF and a declining pattern in the PACF.
Combinations (non-pure ARMA) are more difficult to diagnose.
A single spike in both the ACF and PACF can usually be modelled well by an MA at that lag.
Any declining pattern in the ACF indicates an AR process starting at the lag at which the initial spike appears.

Tip

Always keep in mind that we are moving from a position of ignorance. Significant results in the ACF and PACF may arise purely by chance.

ARIMA PDQ Notation

A univariate time series can be characterised as ARIMA(p, d, q) where:

d is the number of differences
p is the number of AR terms
q is the number of MA parameters

Once we know (p, d, q) for a particular series, we can write it out. For instance, if Y_t is ARIMA (1, 1, 1):

(1-L)Y_t = \phi(1-L)Y_{t-1} + \epsilon_t + \theta\epsilon_{t-1}

Invertibility

The Box-Jenkins approach requires that a variable be stationary and invertible.
This is why we start out determining whether or not the series is stationary.
A time series is said to be “invertible” if it can be represented by a finite-order or convergent autoregressive process.

Filtering Models

Once we think we have the right (p, d, q) model, we test it with a filtering model.

Estimate using MLE and examine the parameter estimates and the ACF, PACF, and Q statistics for the model residuals.
If we see pure white noise in the residuals, we have a good model. If not, re-specify until we do.
Another acid test is the model’s ability to forecast — ex-post forecasting is a good way to settle disputes between rival models.
We can also use AIC and other model selection criteria.
There could be more than one model that provides the desiderata.

Model Selection Criteria

One critique of R^2 is that it can only go up with the addition of new variables. Penalty-based criteria build in parsimony.

Adjusted R^2: \bar{R}^2 = 1 - \frac{n-1}{n-K}(1-R^2)

Akaike Information Criterion (AIC): \text{AIC}(K) = s_y^2(1-R^2)\,e^{2K/n}

Bayesian Information Criterion (BIC / Schwartz criterion): \text{BIC}(K) = s_y^2(1-R^2)\,n^{K/n}

For information criteria: smaller values indicate better-fitting, more parsimonious models. It is often suggested that AIC has better properties for forecasting models while BIC (generally) has a more extreme penalty for each parameter leading to more parsimonious explanations.

Fractional Integration (Discussed in Chapter 7)

Starting from: Y_t - Y_{t-1} = \epsilon_t \;\;\Longrightarrow\;\; (1-L)Y_t = \epsilon_t \;\;\Longrightarrow\;\; (1-L)^1 Y_t = \epsilon_t

We can generalise by allowing d to take non-integer values:

(1-L)^d Y_t = \epsilon_t

Value of d	Process type
-0.5 < d < 0	Anti-persistent, stationary
0 < d < 0.5	Long-memoried, stationary (mean-reverting)
0.5 \leq d < 1	Long-memoried, non-stationary
d = 1	Classic unit root (random walk)

ARFIMA

The ARFIMA model generalises the ARIMA model by allowing non-integer degrees of integration.
ARFIMA models provide a solution for the tendency to over-difference stationary series that exhibit long-run dependence.
In the standard ARIMA approach, d is constrained to integers (0, 1, 2, …). Many series exhibit too much dependence to be I(0) but are not I(1) — ARFIMA captures this middle ground.
Such series are said to be integrated of order d, denoted I(d).

ARFIMA, continued

The standard ARIMA model in lag-operator notation:

(1-L)Y_t = \frac{\theta(L)}{\phi(L)}\epsilon_t

The ARFIMA generalisation:

(1-L)^d Y_t = \frac{\theta(L)}{\phi(L)}\epsilon_t

where d can now range continuously from -0.5 to 1 (and beyond).

Previously we used integer values: d \in \{0, 1, 2\}.
ARFIMA allows d to take values across the full range, with key regions at (-0.5, 0), (0, 0.5), and (0.5, 1).
This is the ARFIMA model, where “FI” stands for Fractional Integration.

A simple special case — the Pure Fractional Noise model:

(1-L)^d Y_t = \epsilon_t, \quad \epsilon_t \sim N(0, \sigma^2)

Lebo, Walker, and Clarke (2000) provide evidence that many political time series are consistent with fractional integration.

Interventions Generally

Interventions or “impact assessments” are a popular extension of ARIMA modeling.
In the most basic models of interventions, we are testing a theory about the impact of a single event on our dependent variable of interest.
If we start out with a stationary dependent variable (Y_t), we can write an intervention model compactly as:

Y_t = f(I_{t-i}) + N_t

where:

I_{t-i} is the intervention component,
i represents your expectation of when the intervention takes effect, and
N_t is the noise model (AR and MA terms).

Interventions Generally (cont.)

Interventions in time series can be characterised by two dimensions:

Dynamics: abrupt vs. gradual
Duration: temporary vs. permanent

Interventions can also be written generally as:

Y_t = \frac{\omega_0}{1 - \delta L}\, I_{t-i} + N_t

where \delta is known as the “adjustment parameter”.

The four types of intervention are determined by the values of I_{t-i} and \delta.
A typical restriction is that |\delta| < 1.
Negative values of \delta are particularly odd and imply model misspecification.
Thus \delta should usually be between 0 and +1.
- Larger values indicate the adjustment takes longer to play out.
If \delta = 0, the intervention is abrupt: impact occurs entirely in the period t - i.

Permanent Interventions

For permanent intervention effects, I_t = 0 before the event and I_t = 1 thereafter.

E.g., I_t: 00000000011111111111

This formulation of I_t is also called a step function.

Abrupt permanent intervention (stationary Y_t, \delta = 0):

Y_t = \omega_0 I_{t-i} + N_t

This is a special case of the general model with \delta = 0.

If we keep the same step-function I_t but set 1 > \delta > 0, we have a gradual permanent intervention.

Temporary Interventions

For temporary intervention effects, I_t = 0 before the event, I_t = 1 at the time of the event, and I_t = 0 thereafter.

E.g., I_t: 00000000010000000000

Abrupt temporary intervention (stationary Y_t, \delta = 0):

Y_t = \omega_0 I_{t-i} + N_t

This is again the general model with \delta = 0.

If we keep the same impulse I_t but set 0 < \delta < 1, we have a gradual temporary intervention.

Understanding Delta

Starting with the general formula for an ARIMA(0,0,0) Y_t (setting i = 0 and ignoring the noise term):

Y_t = \frac{\omega_0}{1 - \delta L}\, I_t

Multiply both sides by (1 - \delta L):

(1 - \delta L)\,Y_t = \omega_0 I_t

Expanding the lag operator:

Y_t - \delta Y_{t-1} = \omega_0 I_t

\boxed{Y_t = \delta\, Y_{t-1} + \omega_0 I_t}

This reveals Y_t as an AR(1) process driven by the intervention: each period’s value depends on the previous period scaled by \delta, plus the current intervention impulse.

Four Intervention Types

Temporary Gradual Intervention: Effect of \delta

Permanent Gradual Intervention: Effect of \delta

Note: For \delta = 1 the step input induces a random walk (no long-run equilibrium).

Alternative Specification of Gradual Interventions

It is also possible to model a gradual intervention while keeping \delta = 0:

Y_t = \omega_0 I_{t-i} + N_t

Instead of estimating \delta, we construct a ramp directly in I_t.

E.g., I_t: 0 0 0 .25 .5 .75 1 0 0 0

Caution: this approach imposes strong metric assumptions about the shape and timing of the transition — assumptions that are baked into the variable construction rather than estimated from the data.

Enders (2004) Figure 5.3: Typical Intervention Patterns

Four canonical shapes (Enders, Applied Econometric Time Series, Figure 5.3):

Panel	Label	Input	\delta	Description
(a)	Pure jump	Step	0	Abrupt permanent
(b)	Pulse	Impulse	0	Abrupt temporary
(c)	Gradually changing	Step	>0	Gradual permanent
(d)	Prolonged pulse	Impulse	>0	Gradual temporary

The table maps directly onto the two-dimensional classification: dynamics (abrupt/gradual) × duration (temporary/permanent).

Steps in a Simple Intervention Model

To model the impact of an intervention we first need a noise model for Y_t — but because we believe an intervention is present, this is potentially problematic.

Enders’ recommendation: Build the noise model using the longer of the two sub-series created by splitting the data at the point of the intervention I_{t-i}.

Alternative approach: Privilege the intervention variable —

Place the intervention on the right-hand side of a model with Y_t made stationary.
Build the noise model from the residuals of that model.

There are multiple valid approaches; the key is not to let the intervention period contaminate the ARIMA identification step.

Differencing with Interventions

So far we have assumed a stationary Y_t. If Y_t is non-stationary and must be differenced, the intervention measure I_t must also be differenced.

Consider a permanent (step) intervention:

(1 - L)\,Y_t = \omega_0 I_t + \epsilon_t

Y_t - Y_{t-1} = \omega_0 I_t + \epsilon_t

Y_t = Y_{t-1} + \omega_0 I_t + \epsilon_t

The problem: we have differenced Y_t but not I_t. This introduces a random walk with drift when we actually want a random walk with a level shift. The intervention must be differenced to maintain consistency with the dependent variable transformation.

Transfer Functions: Overview

Box-Jenkins transfer function models extend the ARIMA framework to models with continuous independent variables.

This approach requires a strict assumption: the independent variable must be truly exogenous.
When this holds, shocks to the independent variable transfer to the dependent variable — hence the name.

In transfer function modelling, you either:

Assume a pure random noise error term for the independent variable, or
Incorporate a filtering model for it as well (not currently available in Stata).

A central challenge: determining the appropriate lag at which to specify each independent variable.

What Is the Appropriate Lag?

Consider the model:

\Delta GOVPOPL_t = \omega_1 FALK_t + \omega_2 EMPXL_{t-i} + N_t

Different theories imply different values of i:

i	Mechanism
0	“Market/me” — informational effect is immediate
1	Media disclosure of economic measures
2	Diffuse social network (“mates”) transmission

Determining the Lag: Two Strategies

Strategy 1 — Theory-driven

Use substantive knowledge about the underlying causal process to set i a priori.

Strategy 2 — Data-driven identification

Let the data guide lag specification through:

Pre-whitening, and
Cross-correlation.

Pre-Whitening

Pre-whitening controls for serial autocorrelation before examining cross-correlations:

Identify and fit the univariate ARIMA model for the independent variable X_t; save the residuals \hat{\alpha}_t.
Apply the same ARIMA filter to the dependent variable Y_t; save those residuals \hat{\beta}_t.
Cross-correlate \hat{\alpha}_t and \hat{\beta}_t to reveal the lag structure free of autocorrelation contamination.

Software note: RATS handles pre-whitening for AR (linear) specifications but not for MA (non-linear) processes. A second-best approach is to filter Y_t using its own univariate ARIMA and then cross-correlate with the pre-whitened X_t.

Cross-Correlation

\Delta GOVPOPL_t = \omega_1 FALK_t + \omega_2 EMPXL_{t-i} + N_t

Suppose we expect the effect of unemployment (EMPXL) to be negative and believe i < 4 (effect materialises within one quarter).

Procedure:

Correlate EMPXL_t with GOVPOPL at lags t, t+1, t+2, t+3.
Select the lag at which the largest negative correlation occurs.
Use that lag as i in the transfer function model.

This data-driven identification strategy is known as cross-correlation.

This complete analysis and the original article can be found on Box

Dynamic Linear Models

The ARIMA approach is fundamentally inductive. The workflow involves the use of empirical values of ACFs and PACFs to engage in model selection. Dynamic models engage theory/structure to impose more stringent assumptions for producing estimates.

Time Series Linear Models/Dynamic Models

First, a result. Aitken Theorem

In a now-classic paper, Aitken generalized the Gauss-Markov theorem to the class of Generalized Least Squares estimators. It is important to note that these are GLS and not FGLS estimators. What is the difference? The two GLS estimators considered by Stimson are not strictly speaking GLS.

Definition \hat{\beta}_{GLS} = (\mathbf{X}^{\prime}\Omega^{-1}\mathbf{X})^{-1}\mathbf{X}^{\prime}\Omega^{-1}\mathbf{y} > Properties >
> (1) GLS is unbiased.
> (2) Consistent.
> (3) Asymptotically normal.
> (4) MV(L)UE

A Quick Example

The variance/covariance matrix of the errors for a first-order autoregressive process is useful to derive.

The matrix is banded; observations separated by one point in time are correlated \rho. Period two is \rho^2; the corners are \rho^{T-1}. The diagonal is one.
What I have actually described is the correlation; the relevant autocovariances are actually defined by \frac{\sigma^{2}\rho^{s}}{1 - \rho^2} where s denotes the time period separation.
It is also straightforward to prove (tediously through induction) that this is invertible; it is square and the determinant is non-zero having assumed that |\rho < 1|.
- The 2x2 determinant is \frac{1}{1-\rho^2}.
- The 3x3 is 1*(1-\rho^2) - \rho(\rho - \rho^3) + \rho^2(\rho^2 - \rho^2). The first term is positive and the second term is non-zero so long as \rho \neq 0. But even if \rho=0, we would have an identity matrix which is invertible.

\Phi = \sigma^{2}\Psi = \sigma^{2}_{e} \left(\begin{array}{ccccc}1 & \rho^{1} & \rho^{2} & \ldots & \rho^{T-1} \\ \rho^1 & 1 & \rho^1 & \ldots & \rho^{T-2} \\ \rho^{2} & \rho^1 & 1 & \ldots & \rho^{T-3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \rho^{T-1} & \rho^{T-2} & \rho^{T-3} & \ldots & 1 \end{array}\right)

given that e_{t} = \rho e_{t-1} + \nu_{t}. A Toeplitz form….

If the variance is stationary, we can rewrite, \sigma^{2}_{e} = \frac{\sigma^{2}_{\nu}}{1 - \rho^{2}}

A comment on characteristic roots….

Cochrane-Orcutt

We have the two key elements to implement this except that we do not know \rho; we will have to estimate it and estimates have uncertainty. But it is important to note this imposes exactly an AR(1). If the process is incorrectly specified, then the optimal properties do not follow. Indeed, the optimal properties also depend on an additional important feature.

What does the feasible do?

We need to estimate things to replace unknown covariance structures and coverage will depend on properties of the estimators of these covariances.

Consistent estimators will work but there is euphemistically considerable variation in the class of consistent estimators.
Contrasting the Beck and Katz/White approach with the GLS approach is a valid difference in philosophies.¹ One takes advantage of OLS and Basus Theorem; one goes full Aitken.

Incremental Models

y_{t} = a_{1} y_{t-1} + \epsilon_{t}

is the simplest dynamic model but it cannot be estimated consistently, in general terms, in the presence of serial correlation. Why?

The key condition for unbiasedness is violated because \mathbb{E}(y_{t-1}\epsilon_{t}) \neq 0. OLS will not generally work.

A note on dynamic interpretation.

Incremental Models with Covariates

y_{t} = a_{1} y_{t-1} + \beta X_t + \epsilon_{t}

The problem is fitting and the key issue is white noise residuals post-estimation. But we have to assume a structure and implement it.

Distributed Lag Models

y_{t} = \alpha + \beta_{0} X_t + \beta_{1}x_{t-1} + \ldots + \epsilon_{t}

The impact of x occurs over multiple periods. It relies on theory, or perhaps analysis using information criteria/F [owing to quasi-nesting and missing data]. OLS is a fine solution to this problem but the search space of models is often large.

In response to this problem, we have structured distributed lag models; there are many such schemes.

Koyck/Geometric decay:
short run and long-run effects are parametrically identified y_t = \alpha + \beta(1-\lambda)\sum_{j=0}^{\infty}\lambda^{j}X_{t-j} + \epsilon
Almon (more arbitrary decay) y_{it} = \sum_{t_{A}=0}^{T_{F}} \rho_{t_{A}}x_{t - t_{A}} + \epsilon_{t} with coefficients that are ordinates of some general polynomial of degree T_{F} >> q. The \rho_{t_{A}} = \sum_{k=0}^{T_{F}} \gamma_{k}t^{k}.

Autoregressive Distributed Lag Models

y_{t} = \alpha + \gamma_{1}y_{t-1} + \beta_{0} X_t + \beta_{1}X_{t-1} + \beta_{2}X_{t-2} + \ldots + \epsilon_{t}

OLS is often used if iid; \epsilon_{t} is unrelated to y_{t-1} is common if nonsensical.
If not iid: GLS is needed.
The authors argue that the lagged dependent variable often yields white noise for free. As they also note, there is a deBoef and Keele paper showing the relationship between these models and a form of error correction models. More on that tomorrow.
There is substance to the timing of impacts.

A Cottage Industry of ADL’s

As recently as April of 2025, a paper appeared in the Journal of Politics advocating the use of ADL(2,2). The paper, by Kagalwala and Whitten, called The Answer was There All Along: Worry about the dynamics!. A previous argument was made for the ADL(1,1) a few years before. I have placed the 2025 paper in the Box.

Structural vs. Non-structural

Data analysis can quite yield models comparisons among competing dynamic structures. The key issue is that the analyst need divine the process; what is the relevant error process and what is the structure and timing of effects alongside the potential question of incremental adjustment. We need good theory for that.

Given such theory, we can take an equations as analysis approach, measure the variables, and derive reduced forms, and then recover parameter estimates deploying simultaneous equations methods. Very large such systems were a core part of early empirical macroeconomics. The failures of such systems led to the proposal of alternatives.

Chris Sims suggested a more flexible approach: the VAR.

VAR: Vector AutoRegression

Choose a relevant set of lag lengths and write each variable in the system as a function of lags of itself and other variables to the chosen lengths.¹ For Stata and for R¹

The key insight is that this VAR is the reduced form to some more complicated as yet unspecified structural form.
But if the goal is to specify how variables related to one another and to use data to discover Granger causality and responses to impulse injected in the system.

A very simple example

library(forecast)
mdeaths
fdeaths
save(mdeaths, fdeaths, file = "./img/LungDeaths.RData")

library(hrbrthemes)
load(url("https://github.com/robertwwalker/Essex-Data/raw/main/LungDeaths.RData"))
Males <- mdeaths; Females <- fdeaths
Lung.Deaths <- cbind(Males, Females) %>% as_tsibble()
Lung.Deaths %>% autoplot() + theme_ipsum_rc()

a VAR

lung_deaths <- cbind(mdeaths, fdeaths) %>%
  as_tsibble(pivot_longer = FALSE)
fit <- lung_deaths %>%
  model(VAR(vars(mdeaths, fdeaths) ~ AR(3)))
report(fit)

Series: mdeaths, fdeaths 
Model: VAR(3) w/ mean 

Coefficients for mdeaths:
      lag(mdeaths,1)  lag(fdeaths,1)  lag(mdeaths,2)  lag(fdeaths,2)
              0.6675          0.8074          0.3677         -1.4540
s.e.          0.3550          0.8347          0.3525          0.8088
      lag(mdeaths,3)  lag(fdeaths,3)  constant
              0.2606         -1.1214  538.7817
s.e.          0.3424          0.8143  137.1047

Coefficients for fdeaths:
      lag(mdeaths,1)  lag(fdeaths,1)  lag(mdeaths,2)  lag(fdeaths,2)
              0.2138          0.4563          0.0937         -0.3984
s.e.          0.1460          0.3434          0.1450          0.3328
      lag(mdeaths,3)  lag(fdeaths,3)  constant
              0.0250          -0.315  202.0027
s.e.          0.1409           0.335   56.4065

Residual covariance matrix:
         mdeaths  fdeaths
mdeaths 58985.95 22747.94
fdeaths 22747.94  9983.95

log likelihood = -812.35
AIC = 1660.69   AICc = 1674.37  BIC = 1700.9

fit2 <- lung_deaths %>%
  model(VAR(vars(mdeaths, fdeaths) ~ AR(2)))
report(fit2)

Series: mdeaths, fdeaths 
Model: VAR(2) w/ mean 

Coefficients for mdeaths:
      lag(mdeaths,1)  lag(fdeaths,1)  lag(mdeaths,2)  lag(fdeaths,2)  constant
              0.9610          0.3340          0.1149         -1.3379  443.8492
s.e.          0.3409          0.8252          0.3410          0.7922  124.4608

Coefficients for fdeaths:
      lag(mdeaths,1)  lag(fdeaths,1)  lag(mdeaths,2)  lag(fdeaths,2)  constant
              0.3391          0.2617         -0.0601         -0.2691  145.0546
s.e.          0.1450          0.3510          0.1450          0.3369   52.9324

Residual covariance matrix:
         mdeaths  fdeaths
mdeaths 62599.51 24942.79
fdeaths 24942.79 11322.70

log likelihood = -833.17
AIC = 1694.35   AICc = 1701.98  BIC = 1725.83

fit %>%
  fabletools::forecast(h=12) %>%
  autoplot(lung_deaths)

Female

lung_deaths %>%
model(VAR(vars(mdeaths, fdeaths) ~ AR(3))) %>%
  residuals() %>% 
  pivot_longer(., cols = c(mdeaths,fdeaths)) %>% 
  filter(name=="fdeaths") %>% 
  as_tsibble(index=index) %>% 
  gg_tsdisplay(plot_type = "partial") + labs(title="Female residuals

Male

lung_deaths %>%
model(VAR(vars(mdeaths, fdeaths) ~ AR(3))) %>%
  residuals() %>% 
  pivot_longer(., cols = c(mdeaths,fdeaths)) %>% 
  filter(name=="mdeaths") %>% 
  as_tsibble(index=index) %>% 
  gg_tsdisplay(plot_type = "partial") + labs(title="Male residuals

Easy Impulse Response

What happens if I shock one of the series; how does it work through the system?

The idea behind an impulse-response is core to counterfactual analysis with time series. What does our future world look like and what predictions arise from it and the model we have deployed?

Whether VARs or dynamic linear models or ADL models, these are key to interpreting a model in the real world.

Males

VARMF <- cbind(Males,Females)
mod1 <- vars::VAR(VARMF, p=3, type="const")
plot(vars::irf(mod1, boot=TRUE, impulse="Males"))

Female

plot(vars::irf(mod1, boot=TRUE, impulse="Females"))

Day 3: Heterogeneity and Dynamics in Data

Outline for Day 3

A Note on Notation from Google AI

The Box-Jenkins ARIMA Approach

Portmanteau Tests for White Noise

Stata’s corrgram Output — An Example

Logging Time Series Variables

Detecting AR and MA Processes

Detecting AR and MA Processes — Rules of Thumb

ARIMA PDQ Notation

Invertibility

Filtering Models

There could be more than one model that provides the desiderata.

Model Selection Criteria

Fractional Integration (Discussed in Chapter 7)

ARFIMA

ARFIMA, continued

Interventions Generally

Interventions Generally (cont.)

Permanent Interventions

Temporary Interventions

Understanding Delta

Four Intervention Types

Temporary Gradual Intervention: Effect of \delta

Permanent Gradual Intervention: Effect of \delta

Alternative Specification of Gradual Interventions

Enders (2004) Figure 5.3: Typical Intervention Patterns

Steps in a Simple Intervention Model

Differencing with Interventions

Transfer Functions: Overview

What Is the Appropriate Lag?

Determining the Lag: Two Strategies

Pre-Whitening

Cross-Correlation

Dynamic Linear Models

Time Series Linear Models/Dynamic Models

A Quick Example

Cochrane-Orcutt

What does the feasible do?

Incremental Models

Incremental Models with Covariates

Distributed Lag Models

Autoregressive Distributed Lag Models

A Cottage Industry of ADL’s

Structural vs. Non-structural

VAR: Vector AutoRegression

A very simple example

a VAR

Female

Male

Easy Impulse Response

Males

Female

Stata’s `corrgram` Output — An Example