Cross-sectional methods assume observations are independent draws from a population. A time series violates that assumption by construction: this month's CPI is last month's plus a change, and the change itself is related to earlier changes. Every method in this course exists because ordinary regression, applied to such data, gives standard errors that are wrong and relationships that are not there.

Every time-series analysis begins with a line plot of the series against time, and most of the mistakes in published applied work could have been avoided by looking at it. The plot shows the trend, the seasonality, the outliers, the breaks, the changes in variance and the missing stretches. It also shows whether the series is one thing or two things spliced (a base-year revision, a definition change) that no test will diagnose.

With autocorrelation ρ at lag 1, the variance of a sample mean is inflated by roughly (1 + ρ) / (1 − ρ). At ρ = 0.8 that factor is 9: ten years of monthly data carry about as much information about the mean as 13 independent observations. Standard errors computed as if the observations were independent are too small by a factor of three, and t-statistics too large by the same.

Assumed: the regression material in Econometrics 101 and the descriptive statistics in Data Analysis 101. Each method is presented with what it assumes, what it produces, how to read the output, and the mistake most often made with it in South Asian applied journals. Worked numbers are illustrative unless a source is named.

A series is thought of as the sum (or the product) of a trend-cycle T_t, a seasonal component S_t and a remainder R_t. The additive form, y_t = T_t + S_t + R_t, suits a series whose seasonal swings are roughly constant in size. The multiplicative form, y_t = T_t × S_t × R_t, suits one whose swings grow with the level, and taking logs turns it into the additive form.

For research use, STL is enough and easier to explain. For a series that will be compared with an official seasonally adjusted one, use X-13 with the same options, or the difference between yours and theirs will be the method, not the data.

The two look alike in a plot of fifty years and call for opposite treatments. Detrending a random walk leaves a non-stationary residual and spurious cycles; differencing a trend-stationary series introduces an MA error and throws away information. Nelson and Plosser (Journal of Monetary Economics 1982, 10:139) tested fourteen US macro series and could not reject a unit root in thirteen, which is why the unit-root test (section 03) sits before every model in this course.

Time-series methods assume regular spacing and no gaps, and administrative data provide neither. A monthly HMIS series has districts that reported late, months with zero because the form was not filed, and a reporting-system change in 2018. Weekly mandi prices have no trades some weeks. The choices made here decide everything after, and they should be listed in the methods.

Stationarity is what makes one history usable: if the process generating the series is the same throughout, the past describes the future and averages over time estimate the process's parameters. Without it, a sample mean is the mean of nothing in particular, regression t-statistics have non-standard distributions, and forecasts inherit whichever part of the past the model happened to see.

y_t = y_t−1 + ε_t. Each period adds a fresh shock and nothing is forgotten, so y_t is the sum of every shock since the start: its variance is tσ² and grows without bound. Plotted, a random walk shows long swings that look like trends and cycles, and no two simulated random walks look alike. Every one of those 'trends' is noise accumulated.

With sixty quarterly observations, the ADF test has perhaps a one-in-three chance of rejecting a unit root when the true autoregressive coefficient is 0.9. A stationary but persistent series will therefore be labelled I(1) most of the time, and the label decides the model. The test also assumes no break: Perron (Econometrica 1989, 57:1361) showed that a stationary series with one shift in its mean or trend is nearly always classified as a unit root by ADF.

Difference until the tests say the series is stationary and no further. One difference removes a stochastic trend; a seasonal difference removes stable seasonality; a second ordinary difference is needed only when the growth rate itself wanders, which is unusual outside high-inflation episodes. Each difference costs observations and, if unnecessary, introduces a non-invertible moving-average term that ARIMA estimation handles badly.

Twenty years of annual data for one state cannot settle whether a series has a unit root. Twenty years for 28 states can, if the states' series share the property. Panel unit-root tests pool the information: Levin, Lin and Chu (Journal of Econometrics 2002, 108:1) assume a common autoregressive coefficient; Im, Pesaran and Shin (2003, 115:53) average individual ADF statistics and allow each state its own; Fisher-type tests combine the individual p-values.

The point of the family is that a stationary series' autocorrelation structure, however complicated, can usually be reproduced with a few parameters, and a model that reproduces the autocorrelation forecasts well. Nothing in the model is about economics; it is a description of the series' memory.

An ARIMA(p, d, q) model is an ARMA(p, q) fitted to the d-th difference of the series. ARIMA(0,1,0) is the random walk; ARIMA(0,1,1) is simple exponential smoothing; ARIMA(1,1,0) is a series whose growth rate is AR(1). The d comes from section 03, p and q from the ACF and PACF of the differenced series or from an information criterion, and the forecasts are integrated back to levels.

The cycle is iterative: a failed check sends you back to identification. Two candidate models that both pass the checks are compared on AIC or BIC (lower is better; BIC penalises parameters more) and, where forecasting is the aim, on out-of-sample accuracy. Report the models that were tried, not only the winner.

Hyndman and Khandakar's algorithm (Journal of Statistical Software 2008, 27(3)), in R's forecast and fable packages and Python's pmdarima, chooses d and D by unit-root tests and then searches over p, q, P, Q by AICc with a stepwise procedure. It is fast, reproducible, and usually lands near what a careful analyst would choose. It is also a starting point, not a verdict.

A known event that hits a series is modelled, not deleted. A pulse dummy (1 in one period) captures a one-off; a step dummy (1 from a date onward) captures a permanent shift; a step passed through the model's dynamics captures a shift that arrives gradually. India's series carry several: November 2016 (demonetisation), July 2017 (GST), April 2020 (lockdown), and base-year revisions that need splicing rather than dummies.

The one-step forecast substitutes the last observed values and the last residuals into the model; the two-step forecast substitutes the one-step forecast, and so on. For a stationary ARMA the forecasts converge to the mean and the interval widens to the unconditional variance; for an I(1) model the forecasts follow the drift and the interval widens without limit, at a rate proportional to √h.

The forecast is a weighted average of past observations with weights that decline geometrically: ŷ_t+1 = αy_t + (1 − α)ŷ_t, with 0 < α < 1. A large α trusts the latest observation; a small one averages over a long memory. α is estimated by minimising the one-step squared errors. The forecast is flat: the same number at every horizon.

Hyndman, Koehler, Ord and Snyder (Forecasting with Exponential Smoothing, Springer, 2008) showed that every smoothing method is the forecast function of a state-space model with Error, Trend and Seasonal components, each Additive, Multiplicative, Damped or None. ETS(A,N,N) is simple smoothing; ETS(A,Ad,M) is damped Holt-Winters with multiplicative seasonality. The state-space form gives likelihoods, so models are chosen by AICc and forecast intervals come from the model rather than from ad hoc formulas.

A forecast of clinic attendance can use rainfall; a forecast of mandi prices can use the sown area. A regression with ARIMA errors, y_t = βx_t + η_t with η_t following an ARIMA process, keeps the regressor's effect and lets the error carry the series' memory. OLS with the same regressor and autocorrelated errors gives the right β and the wrong standard error; the ARIMA errors fix both.

A state programme forecasts caseload for 700 blocks, 38 districts and the state. Forecasting each level separately produces forecasts that do not add up, and the district forecast made from district data ignores what the blocks know. Reconciliation methods (bottom-up, top-down, and the optimal 'MinT' combination of Wickramasuriya, Athanasopoulos and Hyndman, JASA 2019, 114:804) produce forecasts at every level that are coherent and, usually, more accurate than any single level's.

The statistical forecast does not know that a new district hospital opens in March or that the state has announced a scheme. The manager does. The evidence on adjusting model forecasts by judgement (Fildes and colleagues, International Journal of Forecasting 2009, 25:3, on 60,000 adjustments in four firms) is that large adjustments made for a stated reason improve accuracy, small adjustments made from habit worsen it, and optimistic adjustments are the worst.

A model with enough parameters fits any history. R², in-sample RMSE and 'the fitted line tracks the data closely' say nothing about the next twelve months, and models chosen on in-sample fit are systematically overfitted. Accuracy is measured on observations the model did not see: hold back the last portion of the series, fit on the rest, forecast the holdout, compare.

Model A has RMSE 4.1 and model B 4.4 on 36 out-of-sample months. Whether that difference is more than noise is a testable question: Diebold and Mariano (Journal of Business & Economic Statistics 1995, 13:253) test whether the mean of the loss differential, d_t = e_A,t² − e_B,t², is zero, with a standard error that allows the differentials to be autocorrelated (as multi-step errors are).

Model-based intervals account for the error term's variance and nothing else. They ignore parameter uncertainty, model uncertainty, breaks, and the fact that the future contains events the past did not. Empirically, 95% intervals from ARIMA and ETS models cover about 80–90% of outcomes, and macro forecasters' intervals have historically done worse. A calibrated interval is one whose coverage on the evaluation set matches its label.

Granger and Newbold (Journal of Econometrics 1974, 2:111) generated pairs of independent random walks and regressed one on the other. With fifty observations, the t-statistic on the slope exceeded the 5% critical value in about three-quarters of the pairs. R² was often above 0.5. Nothing was related to anything. Phillips (Journal of Econometrics 1986, 33:311) proved why: with I(1) variables the t-statistic does not converge; it grows with the sample, so more data make the problem worse.

Consumption and income; the prices of the same commodity in two markets; short and long interest rates; money and nominal GDP, perhaps. Each pair drifts together because something (a budget constraint, arbitrage, a policy rule) pulls them back when they separate. The combination that is stationary is the long-run relationship, and its coefficient is what a levels regression is trying to estimate.

Pesaran, Shin and Smith (Journal of Applied Econometrics 2001, 16:289) proposed testing for a long-run relationship in a single equation, an autoregressive distributed lag model, without first classifying each regressor as I(0) or I(1). The conditional ECM is estimated and an F-test of the joint significance of the lagged levels is compared against two bounds: the lower assumes all regressors I(0), the upper all I(1).

Most effects in time series arrive over several periods. A finite distributed lag, y_t = Σ_j=0..q β_jx_t−j + ε_t, gives the effect at each lag; the sum Σβ_j is the long-run multiplier. Adding lags of y (an ARDL) lets the effect decay geometrically with few parameters: in y_t = φy_t−1 + βx_t + ε_t, the impact effect is β and the long-run effect is β/(1 − φ).

Sometimes the question is a single coefficient in a regression of stationary variables and the dynamics are a nuisance. Heteroskedasticity-and-autocorrelation-consistent (HAC) standard errors (Newey and West, Econometrica 1987, 55:703) leave the OLS coefficients alone and correct their standard errors for serial correlation up to a chosen lag. The coefficients are unbiased if the model is right; only the inference was wrong.

A cointegrating relationship between money growth and inflation is a statement that they move together in the long run. It does not say which moves which, whether a third thing moves both, or what would happen if the central bank changed money growth on purpose. Those are causal questions, and a time-series regression answers them only under assumptions that must be argued: that x is not itself responding to y, that no omitted series drives both, that the relationship would survive the intervention (the Lucas critique).

Sims (Econometrica 1980, 48:1) proposed treating a small set of macro variables symmetrically: each is regressed on p lags of itself and of all the others. No variable is assumed exogenous; the data decide the dynamics. A VAR with k variables and p lags has k²p slope coefficients, which is why VARs are small: three or four variables, two to four lags, and a sample long enough to estimate forty coefficients.

The name has caused fifty years of confusion. Granger causality is incremental predictive content, and it fails as causation in every direction: a variable that anticipates another (the weather forecast Granger-causes the rain), two effects of a common cause, a variable omitted from the system, and a policy variable set in response to forecasts of the outcome (which will appear to be Granger-caused by it).

An impulse response function traces the effect of a one-time shock to one variable on itself and on the others, period by period, holding other shocks at zero. It is the VAR's answer to 'what happens to inflation after a policy-rate shock, and when?' The difficulty is that the estimated errors are correlated across equations, so a 'shock to the policy rate' has to be separated from the part of the rate's error that is just the economy's own noise arriving in the same quarter.

Both tools answer the same question as the IRF with different trade-offs. Local projections are simpler to explain, more flexible and noisier at long horizons; VAR IRFs are smoother and more dependent on the model being right. A careful paper shows both agree on the shape.

When the variables are I(1) and cointegrated, the VAR in differences is misspecified (it omits the error-correction terms) and the VAR in levels wastes the long-run restrictions. The VECM has both: each variable's change depends on lagged changes of all variables and on the lagged cointegrating residual(s), with a speed-of-adjustment coefficient per equation. The Johansen procedure (section 07) estimates it; the IRFs and decompositions follow as for a VAR, with permanent effects of some shocks allowed.

Financial returns, exchange-rate changes and some commodity prices have a mean that is nearly unpredictable and a variance that is not: large changes follow large changes, calm follows calm. Engle's ARCH (Econometrica 1982, 50:987) let the conditional variance depend on recent squared errors; Bollerslev's GARCH (Journal of Econometrics 1986, 31:307) added its own lag, so that GARCH(1,1), σ²_t = ω + αε²_t−1 + βσ²_t−1, describes most such series with three parameters.

A linear model says a shock has the same effect in a boom and a slump, a rate rise the same effect at 4% inflation and at 9%. Threshold autoregressions (Tong) switch coefficients when an observed variable crosses a threshold; smooth-transition models let the switch be gradual; Markov-switching models (Hamilton 1989) let an unobserved state govern the regime, with the state's probability estimated each period. All need longer samples than linear models and all are harder to explain.

Daily AQI, hourly electricity load, tick-level prices. Everything in the course applies and three things get harder. Seasonality is multiple: hour of day, day of week, annual. Missing values and irregular timestamps are the norm. And the sample is enormous, so everything is 'significant' and the question becomes size, not existence.

Almost every model in this course is a special case of a state-space model: an unobserved state (level, slope, seasonal, regime) evolves by a transition equation, and the observation is the state plus noise. The Kalman filter computes the likelihood and the best estimate of the state at each date; the smoother revises those estimates using the whole sample. ETS, ARIMA, time-varying-parameter regressions and unobserved-components trend models all fit here.

A programme starts at a known date; an outcome is measured repeatedly before and after. The interrupted time series (ITS) design asks whether the level or the trend of the series changed at that date, against the counterfactual that the pre-intervention trend would have continued. Bernal, Cummins and Gasparrini (International Journal of Epidemiology 2017, 46:348) give the tutorial that applied health and policy work now follows; Wagner and colleagues (2002) gave the segmented-regression form.

The single-series ITS cannot separate the programme from anything else that changed at the same time. A comparison series that was exposed to everything except the programme (a neighbouring district, an outcome the programme should not affect, the same outcome in an unexposed group) turns the design into a difference-in-differences in time-series form: the effect is the change in the treated series' level and slope minus the change in the control's. Lopez Bernal, Cummins and Gasparrini (International Journal of Epidemiology 2018, 47:2082) set out the options.

One state adopts a policy; thirty do not. No single control state matches the treated one, but a weighted average of several might: weights chosen so the combination reproduces the treated state's pre-period outcome path and covariates. The post-period gap between the treated series and its synthetic twin is the estimated effect. Abadie, Diamond and Hainmueller (JASA 2010, 105:493) introduced it with California's tobacco law; Abadie (Journal of Economic Literature 2021, 59:391) reviews a decade of use and misuse.

Brodersen and colleagues (Annals of Applied Statistics 2015, 9:247) fit a Bayesian structural time-series model to the outcome in the pre-period, using control series as regressors, then forecast it through the post-period as the counterfactual. The effect is observed minus forecast, with a credible interval that widens with the horizon. It is an ITS whose counterfactual is a state-space forecast rather than a straight line, and a synthetic control whose weights are regression coefficients rather than constrained to a simplex.

Time-series results are unusually sensitive to small decisions (lag length, deterministic terms, the sample start, a dummy), and the only defence is that every decision is in a script a reader can change. Quarto or R Markdown, with the data download at the top and the tables generated at the bottom, is the form that makes this automatic.