Sampling Interval: When a variable is measured sequentially in time over or at a fixed interval.
Discrete-time stochastic process: A sequence of random variables defined at fixed sampling intervals.
Serially dependent: Observations close together in time tend to be correlated
Seasonal variation: A repeating pattern within each year, although the term is applied more generally to repeating patterns within any fixed period, such as restaurant bookings on different days of the week
Multiple time series: Multiple time series of different variables.
Stochastic trends: Trends that seem to change direction at unpredictable times rather than displaying relatively consistent patterns.
Random walk: A mathematical model to handled Stochastic trends data.
Price index: Is the ratio of the cost of a basket of goods now to its cost in some base year.
Chapter 2
Asymptotically: As the sample size approaches infinity.
Ensemble mean: The average taken across all the possible time series that might have been produced.
Ergodic: A time series model that is stationary in the mean if the time average for a single time series tends to the ensemble mean as the length of the time series increases.
Second-order properties: Include the mean, variance, and serial correlation.
Lag: The number of time steps between the variables.
Autocorrelation: A correlation of a variable with itself at different times. Also known as serial correlation.
Autocovariance function (acvf):\(\gamma_{k} = E[(x_t - \mu)(x_{t+k} - \mu)]\)
Autocorrelation function (acf):\(\rho_k = \frac{\gamma_{k}}{\sigma^2}\)
Sample acf:\(r_k = \frac{c_k}{c_0}\)
Correlogram: a plot of \(r_k\) against k. The x-axis gives the lag (k) and the y-axis gives the autocorrelation (\(r_k\)) at each lag.
Chapter 3
Leading variables:
Cross-correlation function:
Cross-correlgoram:
Second-order stationary:
Exponential smoothing:
Holt Winters:
Chapter 4
Autogressive process:
White noise: Used to refer to series that contained all frequencies in equal proportions, analogous to white light.
Purely random: Another tearm for white noise.
Random walk: A fundamental non-stationary model based on discrete white noise
Discrete white noise: A times series with variables \(w_1, w_2, . . . , w_n\) that are independent and identically distributed with a mean of zero.
Synthetic series: Time series simulated using a model.
Bootstrapping: Simulations to generate plausible future scenarios and to construct confidence intervals for model parameters.
Chapter 5
Serially correlated:
Generalized Least Squares: The procedure is essentially based on maximising the likelihood given the autocorrelation in the data and is implemented in R in the gls() function (within the nlme library). _ Time Series Forecast: Think of a forecast from a regression model as an expected value conditional on past trends continuing into the future.