Univariate Time Series Analysis

Kevin Sheppard

Forecasting

• One-step forecast

$$ E_t[Y_{t+1}] $$

• Multi-step forecast

$$ E_t[Y_{t+h}] $$

• Forecasts are recursively computed

  • First compute $E_t[Y_{t+1}] $ which depends on the $\mathcal{F}_t$ information $Y_t, Y_{t-1}, \ldots, \hat{\epsilon}_{t}, \hat{\epsilon}_{t-1} \ldots$
  • Then compute $E_t[Y_{t+2}] $ which depends on $E_t[Y_{t+1}]$ and $\mathcal{F}_t$ information
  • Repeat until $E_t[Y_{t+h}] $

• Forecasts from covariance stationary time series are mean reverting

AR(1) Model Fitting

$$ Y_t = \phi_0 + \phi_1 Y_{t-1} + \epsilon_t $$

res = SARIMAX(m2_growth, order=(1, 0, 0), trend="c").fit()
summary(res)

	coef	std err	z	P>|z|	[0.025	0.975]
intercept	0.0021	0.000	11.315	0.000	0.002	0.002
ar.L1	0.6326	0.010	62.755	0.000	0.613	0.652
sigma2	1.221e-05	1.85e-07	65.856	0.000	1.18e-05	1.26e-05

Forecasts

$$ \mathrm{E}_t[Y_{t+h}] = \phi_0 + \phi_1 \mathrm{E}_t[Y_{t+h-1}] + \mathrm{E}_t[\epsilon_{t+h}] $$

fcast = res.forecast(12)
plot(fcast, y=-2)

Manually Constructing Forecasts

• Forecasts are directly computes using the recursive form

$$ \mathrm{E}_t[Y_{t+h}] = \phi_0 + \phi_1 \mathrm{E}_t[Y_{t+h-1}] + \mathrm{E}_t[\epsilon_{t+h}] $$

• Rules for computing $E_t[Y_{t+h}]$
  1. $\mathrm{E}_t[\epsilon_{t+h}]=0$ if $h>0$
  2. $\mathrm{E}_t[\epsilon_{t+h}]=\hat{\epsilon}_{t+h}$ if $h\leq 0$
  3. $\mathrm{E}_t[Y_{t+h}]=Y_{t+h}$ if $h\leq 0$
  4. Replace $\mathrm{E}_t[Y_{t+h-j}]$ with its value previously computed

p0 = res.params["intercept"]
p1 = res.params["ar.L1"]
direct = pd.Series(np.zeros(12), index=fcast.index)
for i in range(12):
    if i == 0:
        direct[i] = p0 + p1 * m2_growth.iloc[-1]
    else:
        direct[i] = p0 + p1 * direct[i - 1]

Manual Forecasts

plot(direct)

Recursive Forecast Construction

• Simple to construct forecasts recursively

t = m2_growth.shape[0]
half = t // 2

forecasts = []
res = SARIMAX(m2_growth.iloc[:half], order=(1, 0, 0), trend="c").fit()
forecasts.append(res.forecast(1))
for i in range(half, t):
    res = res.extend(m2_growth.iloc[i : i + 1])
    forecasts.append(res.forecast(1))
h1 = pd.concat(forecasts)
h1.name = "h1"

Fast one-step forecast construction

• extend using second half and use get_prediction()

res2 = SARIMAX(m2_growth.iloc[:half], order=(1, 0, 0), trend="c").fit()
res2 = res2.extend(m2_growth.iloc[half:])
fast_h1 = res2.get_prediction().predicted_mean
fast_h1.name = "fast_h1"
pd.concat([h1, fast_h1], axis=1).head()

	h1	fast_h1
1990-06-01	0.001715	0.001715
1990-07-01	0.004789	0.004789
1990-08-01	0.004279	0.004279
1990-09-01	0.005718	0.005718
1990-10-01	0.004645	0.004645

Forecasts and Realizations

fig, ax = plt.subplots(1, 1)
h1.plot(ax=ax)
_ = realizations.plot(ax=ax)

Forecast Errors

• 1-step should be a $WN$ process
• $h$-step should be at most a $MA(h-1)$

fe = (realizations - h1).dropna()
plot(fe, 13)

Forecast Error Autocorrelation

• 1-step should be a $WN$ process
• $h$-step should be at most a $MA(h-1)$

acf_pacf_plot(fe, 24, size=-2)

Multi-step Forecasts

• Multi-step forecasts are recursively generated

multistep_forecast_plot()