The past, the present, and the future
Levy Economics Institute
September 18, 2024
You are considering investing in a company stock, and you want to know how risky that investment is. You have downloaded data on daily stock prices for many years. How should you define returns? How should you assess whether and to what extent returns on the company stock move together with market returns?
Heating and cooling are important uses of electricity. How does weather conditions affect electricity consumption? We are going to use monthly data on temperature and residential electricity consumption in Arizona. How to formulate a model which captures these factors?
tsset commandRegressions: to condition \(y_t\) on values of \(x_t\) the two variables need to be on the same frequency. Otherwise, we need to adjust one of them.
How to adjust? Aggregation!
Time series regressions are special for several reasons…but many aspects remain the same
Time series data “tends” to have trends. how to detect them?.
Define change (or fist difference): \(\Delta y_t = y_t - y_{t-1}\)
\[ \begin{align} \text{Positive trend}: & E[\Delta y_t] > 0 \\ \text{Negative trend}: & E[\Delta y_t] < 0 \end{align} \]
\[ \begin{align} \text{Linear trend}: & E[\Delta y_t] = \text{constant} \\ \text{Exponential trend}: & E[\Delta \ln(y_t)] = \text{constant} \end{align} \]
To learn something, we need consistency in patterns If the patterns change constantly, we cannot learn anything from the data
In absence of stationarity, we cannot learn anything from the data Or what we learn may be misleading
Some non-stationary variables are not easy to deal with:
We can test if a series is a random walk.
Phillips-Perron test is based on this model: \[y_t = \alpha + \rho y_{t-1} + e_t\]
This model represents a random walk if \(\rho = 1\), which is also called a unit root
Random walk test = testing if the series has a unit root.
The Phillips-Perron test has hypothesis
\[H_0: \rho = 1 \text{ vs } H_A: \rho < 1\]
This test does not follow the usual t-distribution. Has its own distribution.
in Stata, you can use it with the pperron command. (see helpfile)
When the p-value is large (e.g., larger than 0.05), we don’t reject the null, concluding that the time series variable follows a random walk
Many versions of unit root test.
\[\Delta y_t = \alpha + \delta y_{t-1} + e_t\]
dfuller
Key decisions:
qui: {
use "data_slides/stock-prices-daily.dta", clear
ren *, low
sort ym date
bysort ym: gen flag = _n == _N
keep if flag==1
tsset ym
gen ret_sp500 = 100 * (p_sp500 - p_sp500[_n-1]) / p_sp500[_n-1]
gen ret_msft = 100 * (p_msft - p_msft[_n-1]) / p_msft[_n-1]
}
line ret_sp500 ym, name(m1, replace)
line ret_msft ym, name(m2, replace)
Phillips–Perron test for unit root Number of obs = 251
Variable: ret_sp500 Newey–West lags = 4
H0: Random walk without drift, d = 0
Dickey–Fuller
Test -------- critical value ---------
statistic 1% 5% 10%
--------------------------------------------------------------
Z(rho) -230.845 -20.301 -14.000 -11.200
Z(t) -14.365 -3.460 -2.880 -2.570
--------------------------------------------------------------
MacKinnon approximate p-value for Z(t) = 0.0000.
Phillips–Perron test for unit root Number of obs = 251
Variable: ret_msft Newey–West lags = 4
H0: Random walk without drift, d = 0
Dickey–Fuller
Test -------- critical value ---------
statistic 1% 5% 10%
--------------------------------------------------------------
Z(rho) -284.851 -20.301 -14.000 -11.200
Z(t) -19.346 -3.460 -2.880 -2.570
--------------------------------------------------------------
MacKinnon approximate p-value for Z(t) = 0.0000.\[y^E_t = \beta_0 + \beta_1 x_{1t} + \beta_2 x_{2t} + \ldots\]
\[\Delta x_t = x_t - x_{t-1}\]
The regression in changes is \[\Delta y^E_t = \alpha + \beta \Delta x_t\]
\[\%\Delta (y_t)^E = \alpha + \beta \%\Delta(x_t)\]
\[\Delta \ln(y_t) = \ln(y_t) - \ln(y_{t1})\]
\[\%\Delta(\text{MSFT}_t) = \alpha + \beta \%\Delta (\text{SP500}_t)\]
\[\Delta y^E_t = \alpha + \beta \Delta x_t\]
\[y^E_t = \alpha + \beta x_t + \delta_{Jan} + \delta_{Feb} + \cdots + \delta_{Nov}\]
Serial correlation means correlation of a variable with its previous values
Order serial correlation coefficient is defined as \[\rho_k = \text{Corr}[x_t, x_{t-k}]\]
If independent variables are serially correlated, usually not a problem.
If dependent variable is serially correlated (the error), it is a problem.
Serial correlation makes the usual standard error estimates wrong.
More precisely it is serial correlation in residuals, but think about is as serial correlation in \(y_t\) is okay
In most time series, there will be some serial correlation
Sol1: Use the Newey-West SE: in Stata newey command.
Sol2: Have lagged dependent variable in the regression (one lag is usually enough) \[y_t = \alpha + \beta x_t + \gamma_1 y_{t-1} + \gamma_2 y_{t-2} \ldots\]
| Variables | (1) | (2) |
|---|---|---|
| \(\Delta\)CD | 0.031** | 0.017** |
| (0.001) | (0.002) | |
| \(\Delta\)HD | 0.037** | 0.014** |
| (0.003) | (0.003) | |
| month = 2, February | -0.274** | |
| month = 3, March | -0.122** | |
| … | ||
| Constant | 0.001 | 0.092** |
| (0.002) | (0.013) | |
| Observations | 203 | 203 |
| Variables | (1) | (2) | (3) |
|---|---|---|---|
| Simple SE | Newey–West SE | Lagged dep.var | |
| \(\Delta\)CD | 0.017** | 0.017** | 0.017** |
| (0.002) | (0.002) | (0.002) | |
| \(\Delta\)HD | 0.014** | 0.014** | 0.014** |
| (0.002) | (0.003) | (0.002) | |
| Lag of \(\Delta\) ln Q | -0.002 | ||
| (0.062) | |||
| Month dummies | YES | YES | YES |
| Observations | 203 | 203 | 202 |
| R-squared | 0.951 | 0.951 | |
| Standard errors in parentheses | |||
| ** p<0.01, * p<0.05 |
In time series, we can analyze how an impact builds up across several periods (time-to-build):
\[\Delta y^E_t = \alpha + \beta_0 \Delta x_t + \beta_1 \Delta x_{t-1} + \beta_2 \Delta x_{t-2}\]
\(\beta_0\) = how many units more \(y\) is expected to change within the same time period when \(x\) changes by one more unit (No change before or after).
\(\beta_1\) = how much more \(y\) is expected to change in the next time period after \(x\) changed by one more unit.
Cumulative effect: \(\beta_\text{cumul} = \beta_0 + \beta_1 + \beta_2\)
\(\beta_k\) are the short-term effects, \(\beta_\text{cumul}\) is the cumulative effect/Long Term Effect.
\[ \begin{aligned} \Delta y^E_t &= \alpha + \beta_0 \Delta x_t + \beta_1 \Delta x_{t-1} + \beta_2 \Delta x_{t-2} \\ \Delta y^E_t &= \alpha + (\beta_\text{cumul} - \beta_1 - \beta_2) \Delta x_t + \beta_1 \Delta x_{t-1} + \beta_2 \Delta x_{t-2} \\ \Delta y^E_t &= \alpha + \beta_\text{cumul} \Delta x_t + \beta_1 (\Delta x_{t-1}- \Delta x_t) + \beta_2 (\Delta x_{t-2} - \Delta x_t)\\ \Delta y^E_t &= \alpha + \beta_\text{cumul} \Delta x_{t} + \delta_0 \Delta(\Delta x_t) + \delta_1 \Delta(\Delta x_{t-1}) \end{aligned}\]
See Case Study in Chapter 12
d.).newey or lagged dependent variable.Rios-Avila and Cia