STT 53 Time Series Analysis Lecture Notes PDF

# STT 53 Time Series Analysis ## Introduction (Statistical Modeling) - aims at provide more flexible tools for data analysis ### Objectives for Modeling a Structure of a System - Understanding and description of the generating mechanism - Forecasting of future values - Optimal control of a sy...

# STT 53 Time Series Analysis ## Introduction (Statistical Modeling) - aims at provide more flexible tools for data analysis ### Objectives for Modeling a Structure of a System - Understanding and description of the generating mechanism - Forecasting of future values - Optimal control of a system ## Statistical Modeling - **Time Series Modeling** - describes how a particular value is influenced by its past values. - **Spatial Modeling** - describes how a particular value is influenced by its "neighboring" value - **Space-Time Modeling** - describes how a particular value is influenced by its "neighboring" and its past value. ## Modeling - **Process** (the set of possible observations on a time-sequenced variable, including a stochastic generating mechanism specifying how the Y's are related through time) - **Realization** (the available data or the time series) - **Model** (a representation of the process, developed by analyzing the realization) ## Common directions - to study the dynamic structure of a process - to investigate the dynamic relationship between variables - to perform seasonal adjustment of economic data - to improve regression analysis when the errors are serially correlated ## Time Series - In forecasting a future value arises from a certain process, time is an important factor that must be considered in our models. - simply a series of data points ordered in time. In a time series, time is often the independent variable and the goal is usually to make a forecast for the future. ## Definition ### Z(w, t): Stochastic Process - Sample space - Indexed set Consider indexed set corresponds to moments of time - For a fixed t, Z(w, t) is a random variable - For a given w, Z(w, t) is called a sample function or a realization as a function of t. ## Time Series - a sequence of values of some variables taken at successively equally spaced time periods like a day, a week, a quarter, a month, or a year. - We assumed data are measured at equally spaced, discrete time intervals - a set of observations generated sequentially in time. Hence, they are dependent (correlated) to each other ## Cross section Data - a sequence of values of some variables taken for a specific period of time and for different entities. - Usually uncorrelated observations are observed in cross section data. ## Approach - **Time-Domain Approach** - views the investigation of lagged relationships as most important - uses ACF, PACF, IACF - **Frequency-Domain Approach** - views the investigation of cycles as most important - uses frequency of occurrence rather than time of occurrence. ## Time Series Components - **Trend** - refers to the long term upward or downward movement that characterizes a time series over a period of time. - reflects the long run growth or decline in the time series. - for example, possible causes of sales movement are increase in total population, market growth or changes in consumer tastes. - not necessarily linear and can be estimated by performing a linear or nonlinear regression of the seasonally adjusted series on time. - **Season** - Regular periodic fluctuations that occur within year. - results from events that are periodic and recurrent. (e.g. monthly changes recurring each year) - **Cycle** - undulating wave-like change around the trend. - refers to up and down fluctuations that are observable over extended period of time. - covers longer period than the seasonal variation. (clear only for series of 20 years or more) - possible cause: change in economic conditions. - difficult to forecast because they are recurrent but not periodic. - Cycles are often irregular both in height of peak and duration - **Irregular** - unpredictable random variation in the time series that the other three components (trend, cycle, seasonality) fail to account for. - possible causes: unforeseen events such as strikes, typhoons, coup d'etat - short duration; unrepeating. - Unpredictable, random, “residual" fluctuations - "Noise" in the time series ## Time Series Decomposition - One method of describing a time series - Decompose the series into various components - In classical decomposition, a time series is described as: $Y_t = f(T_t, C_t, S_t, I_t)$ where: - $Y_t$ is the actual value at time t; - f is a mathematical function; - $T_t$ is the trend component; - $C_t$ is the cyclical influences; ## Two General types of Decomposition - **Additive Model** - use when the peak of the seasonality is constant. $Y_t = T_t + C_t + S_t + I_t$ - used when the variations around the trend do not vary with the level of the time series - **Multiplicative Model** - use when the peak of the seasonality increases. $Y_t = T_t \times C_t \times S_t \times I_t$ - used when the trend is proportional to the level of the time series ## Trend-Cycle Component - trend and cyclical components are grouped into one - **Additive Model** $Y_t = T_tCt + S_t + I_t$ - **Multiplicative Model** $Y_t = T_t Ct \times St \times It$ - **Apply simple moving average to estimate the trend at time t** $T_t = \frac{1}{2a + 1} \sum_{k=-a}^{a} Y_{t+k}$ - **For additive model:** - detrend = time series - $T_t$ - **For multiplicative model:** - detrend = time series / $T_t$ # Decomposition (Season) - Take the average of detrend values for every period (frequency or units of season. e.g monthly, quarter, etc) - This gives an estimate of the seasonal factor $Season_i, i = 1, ..., F$, where F is the length of the season. - For example, monthly data (frequency = 12), at t = 2 (say Year zero, month 2) the $S_t$ is given by: $S_2 = Season_2$ ## Decomposition (Irregular) - **For additive model:** $Irregular_t = detrend_t - S_t$ - **For multiplicative:** $Irregular _t = \frac {detrend_t } {S_t}$ ## Example - **Model: Additive Model** - **Trend** - **detrend** - **Season** - **Irregular**