M.6.1 Econometrics & Time Series from Zero

Basic Prerequisites: Probability & Measure Theory (conditional expectation) - nothing else is assumed.

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1. Intuition & Practical Objective

This page builds the why of time-series econometrics with no prior time-series knowledge needed. The objective is one idea: financial data is a sequence of observations indexed by time, and almost every textbook-statistics habit you bring to it is wrong - so the entire discipline is about learning to ask the right conditional question and about making the data stationary before trusting any inference on it.

Start with the dumbest question: why do we study returns and not prices? Three reasons (Campbell–Lo–MacKinlay, Tsay Ch 1):

1. Scale-free and complete. For an investor the price level is arbitrary - what matters is the percentage change. $P_t=110$ after $P_{t-1}=100$ is the same investment experience as $11$ after $10$. Returns strip out the scale.
2. Better statistical properties. Log returns are approximately normal-ish, near-independent, and (crucially) their variance is roughly stable over time - whereas prices have a mean and variance that drift explosively.
3. Additivity. A $k$-period continuously-compounded log return is the sum of one-period log returns, $r_t[k]=\ln (P_t/P_{t-k})={\sum }_{j=0}^{k-1}{r}_{t-j}$ - which lets the central limit theorem and all of linear-model machinery in. Simple returns don’t add this way.

The second idea is stationarity. A process is the object of study, and the question “can I learn anything stable from the past to forecast the future?” has a precise answer only if the law of the process doesn’t drift with calendar time. That is stationarity: the joint distribution (or just the first two moments) is time-invariant. A random walk is not stationary - its variance grows with $t$. Returns, in the simplest view, are approximately stationary - which is why returns are the unit of analysis.

The third idea, previewed here and built in 04 · Volatility Modeling: even returns are not independent - their volatility clusters. Calm periods follow calm periods; storms follow storms. That non-independence in the second moment is the single most exploitable fact in all of quantitative finance, and it is invisible if you only ever ask “is the mean constant?“

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2. Mathematical Ground Truth & Derivations

Return definitions (Tsay Ch 1, eq. 1.1–1.7).

Simple one-period return: $R_t=\frac{P_t}{P_{t-1}}-1$. Continuously-compounded (log) return: $r_t=\ln (1+R_t)=p_t-p_{t-1}$, where $p_t=\ln P_t$. Multi-period log return = sum of one-period log returns:

$$r_t[k]=p_t-p_{t-k}=\sum _{j=0}^{k-1}{r}_{t-j}.$$

Portfolio simple return is the weighted average of constituents’ simple returns (log returns are only approximately so - Tsay eq. 1.7).

Why the log. Over one step $\ln (P_t/P_{t-1})\approx R_t$ for small moves, but over many steps log returns exactly add while simple returns multiply. And under geometric Brownian motion (see Stochastic Calculus), $p_t$ is Gaussian, so returns are lognormal-consistent.

Random walk. The simplest non-stationary process: $p_t=p_{t-1}+a_t$, i.e. $p_t=p_0+{\sum }_{i=1}^t{a}_i$. Then $\mathrm{Var}(p_t)=t{\sigma }_a^2$ - variance grows linearly with time. A shock never decays. This is the unit root, and it is why prices can never be treated as stationary.

Weak (covariance) stationarity (Tsay §2.1). A series $\{x_t\}$ is weakly stationary iff:

1. $\mathbb{E}[x_t]=\mu <\infty$ - constant, finite mean;
2. $\mathrm{Var}(x_t)={\sigma }^2<\infty$ - constant, finite variance;
3. $\mathrm{Cov}(x_t,x_{t-k})=\gamma (k)$ depends only on the lag $k$, not on $t$.

Strict stationarity requires the entire joint distribution to be time-invariant. For Gaussian processes the two coincide (Tsay §2.1). The autocorrelation function (ACF) is

$${\rho }_k=\frac{\gamma (k)}{\gamma (0)}.$$

For a weakly stationary series the ACF is a function of lag only - which is precisely the object all of linear time-series modeling studies.

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3. Computational Implementation - see stationarity with your own eyes

Simulate a geometric random walk for a “price”, then compare the ACF of the log-price against the ACF of the log-returns. The price ACF hugs 1 (non-stationary - never mean-reverts), the return ACF is ≈0 (stationary - no exploitable linear memory). Stdlib only.

The price’s first autocorrelation is ~0.99 - a shock essentially never decays (unit root). The returns’ ACF at lags 1 and 5 is ~0 - the log return is (approximately) stationary white noise. That single comparison is the entire motivation for differencing. The small positive mean (0.66 bps/step) is the drift; note how tiny it is next to the 2%-per-step noise.

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4. Failure Modes & First-Principles Breakdowns

1. Regressing on levels = spurious regression. If you run $P_t^{(1)}$ on $P_t^{(2)}$ for two independent random walks, you get a high $R^2$ and a huge $t$-statistic - pure artifact of non-stationarity, zero economic content. This is the number-one trap, developed fully in 03 · Forecasting & Unit Roots.
2. Log vs simple return confusions are minor - for the mean. For daily/monthly data the two are nearly equal, but for portfolios simple returns average correctly while log returns do not. Never average log returns of a portfolio.
3. “Stationary” is not “iid.” White noise (uncorrelated) is not independent: GARCH innovations are uncorrelated but dependent (the squared values are autocorrelated). Missing this confuses the mean model with the variance model - the gateway error into 04 · Volatility Modeling.
4. The $\frac{1}{2}{\sigma }^2$ correction. Under GBM, $\ln P_t$ has drift $\mu -\frac{1}{2}{\sigma }^2$, not $\mu$ (Itô’s lemma, Stochastic Calculus). Pricing and long-horizon return forecasts that omit it are biased - see the BSM derivations.

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5. References

• Tsay, Analysis of Financial Time Series
• Campbell, Lo & MacKinlay, The Econometrics of Financial Markets
• Glasserman, Monte Carlo Methods in Financial Engineering

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6. Connected Graph Bridges

• Base: Probability & Measure Theory · Stochastic Calculus & Itô
• Continue: 02 · Stationarity & ARMA · Index Hub