Regression with Time Series Data Week 10
Main features of Time series Data
Observations have temporal ordering
Variables may have serial correlation, trends and seasonality Time series data are not a random sample because the observations in time series are collected from the same objects at different points in time For time series data, because MLR2 does not hold, the inference tools are valid under a set of strong assumptions (TS1-6) for finite samples While TS3-6 are often too restrictive, they can be relaxed for large samples. In essence, TS1, TS2, (z10), (h10) and (u10) are sufficient for large sample inference Serial correlation of a time series variable is the correlation between the variable at one point in time with the same variable at another point in time (z10), (h10), (u10)

z10 = E(ut | xt) = 0
When (z10) holds then the regressors are contemporaneously exogenous and OLS is consistent but is not sufficient for OLS to be unbiased When TS3 holds, which implies (z10), then the regressors are strictly exogenous and OLS is unbiased h10 = Var(ut | xt) = 2 and is known as contemporaneous homoskedasticity and is a weaker assumption than TS4 u10 = E(utus | xt,xs) = 0 and is a weaker assumption than TS5 FDL model and LRP

A FDL model allows the lags of one or more variables to affect the dependent variable The LRP is the impact on y of a permanent one unit shift in z at t Trends and Seasonality
Seasonal dummy variables can be used to account for seasonality with the first quarter as base and define three dummy variables Trends can be accounted for by adding a time trend in the model Estimate B1 and B2 by regressing yt, xt1, xt2 on the time trend and seasonal dummies and save the residuals respectively. Regressing the residual of yt on the two residuals of xt1 and xt2 without intercept. Further Issues with Time Series Data Summary Week 11

1. Asymptotic properties of OLS with time series
Theorem 11.1 (consistency)
Under TS1-2 and (z10), if {xt,yt} are...

...TimeSeries behaviour of BOT in India: Evidence from Co integration Analysis and Error Correction Model
xxxxxxxxxxxxxxxx
Assistant Professor, Department of Business Administration,
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West Bengal University of technology
Kolkata, India
Tel: +91-9231058348 E-mail: partha.s.sarkar@gmail.com
Abstract
India, a developing economy contains trade deficit from its very inception. The main objective of the study is to portray some characteristics of India’s trade in pre liberalization (1951-1991) and post liberalization (1991-2008) period walk in total export and import timeseries. Author attempt to understand the timeseries behavior of total export and import of India. Unit root tests recognize the existence of random walk. Johansen co integration test reveals long-run equilibrium relationship between these two variables. Getting the existence of co integration, the study attempts to find a causal relationship by using error correction mechanism. Additionally Unit root test is employed to examine the existence of stationarity in the given timeseries. Ultimate Test results expose bidirectional long term causality and unidirectional short term causality between imports - exports of India. Findings of the study corroborate that India is unable to violate of its international budget constraints.
Keywords: Stationary, Co integration, Export,...

....2.3 Timeseries models
Timeseries is an ordered sequence of values of a variable at equally spaced time intervals. Timeseries occur frequently when looking at industrial data. The essential difference between modeling data via timeseries methods and the other methods is that Timeseries analysis accounts for the fact that data points taken over time may have an internal structure such as autocorrelation, trend or seasonal variation that should be accounted for. A Time-series model explains a variable with regard to its own past and a random disturbance term. Special attention is paid to exploring the historic trends and patterns (such as seasonality) of the timeseries involved, and to predict the future of this series based on the trends and patterns identified in the model. Since timeseries models only require historical observations of a variable, it is less costly in data collection and model estimation.
. Timeseries models can broadly be categorized into linear and nonlinear Models. Linea models depend linearly on previous data points. They include the autoregressive (AR) models, the integrated (I) models, and the moving average (MA) models. The general autoregressive...

...Forecast of Remittance in Bangladesh
A TimeSeries Forecast
8/11/2012 North South University
Prepared by: Athena Rahmetullah Leonora Adhikari Nudrat Faria Shreya Sumaita Maisha Tajkia Mahmud
I.
INTRODUCTION
Remittances are funds transferred from migrants to their home country. They are the private savings of workers and families that are spent in the home country for food, clothing and other expenditures, and which drive the home economy. Remittance inflows in the economy of Bangladesh are getting larger every passing year, matching with the increasing external demand for its manpower. Remittances have helped improve the social and economic indicators like nutrition, living condition and housing, education, health care, poverty reduction, social security, and investment activities of the recipient households. The relative weight of remittances has also increased against most of the macroeconomic variables alongside the contribution to GDP. Moreover, Bangladesh has been able to avoid any serious imbalances in balance of payments current account, although it has persistent merchandize trade deficits. Not only that, the export tradable sector has thus far remained unaffected from the Dutch Disease effects of remittances. (Dutch Disease effect of remittances is the appreciation of home currency due to increase in remittances.) Remittance inflows in recent years have been instrumental in maintaining the current account surplus despite...

...A timeseries analysis for Chinese Electricity Demand
1. Introduction
Electricity is the basic demand for peoples’ daily life, and relate to the Industrial production. It is also be a very important indexer to indicate the economic growth because the electricity demand and the economic growth always highly related. Thus the prediction of the electricity demand is very important. The government of a country must be able to forecasting the electricity demand in order to formulate its policies. This paper will conduct an analysis for the Chinese Electricity demand, and provide some useful model to predict the electricity demand in the future.
2. Timeseries data for Electricity demand of China
Monthly electricity generation of China from 1999 to 2004 is shown in table 1. Plot the data on Figure 1, it can be shown that the demand of electricity was growing during these six years. From the chart, the growth trend can be easily indentified. The pattern of the trend will be discussed in following section. The monthly electricity data present some seasonal changed pattern, the July and August seem like the peak of each year, more detail should be discussed in later section.
Table 1: Chinese Monthly Electricity Generation
Source: Chinese yearly statistical Data, www.stats.gov.cn
Figure 1: Electricity Generation plot versus time
3. Modelling trend by using Polynomial Functions
From the plot...

...An Introduction to univariate
financial timeseries analysis
1 Introduction: what is a time-series?
Time-series is a sequence
{x1, x2, ..., xT } or {xt} , t = 1, ..., T,
where t is an index denoting the period in time in which x occurs. We
shall treat xt as a random variable; hence, a time-series is a sequence
of random variables ordered intime. Such a sequence is known as a
stochastic process. The probability structure of a sequence of random
variables is determined by the joint distribution of a stochastic process.
A possible probability model for such a joint distribution is:
xt = α + t, t ∼ n.i.d.
¡
0, σ2 ¢
,
i.e., xt is normally independently distributed over time with constant
variance and mean equal to α. In other words, xt is the sum of a constant
and a white-noise process. If a white-noise process were a proper model
for financial time-series, forecasting would not be very interesting as
the best forecast for the moments of the relevant timeseries would be
their uncoditional moments. However this is certainly not the case for
all financial timeseries. Consider the dataset STOCKINT.XLS which
contains, in Excel format, retrieved from Datastream, quarterly timeseries
data for stock index and valuation...

...about a market. These methods are most appropriate when there is not much historical data to work with.
2. Causal methods assume that demand is strongly related to a particular cause, such as environmental or market factors.
3. Timeseries methods are based on the assumption that historical patterns of demand are a good indicator of future demand, and that over a period of time, demand can be charter in three different ways: as an underlying trend (flat, up , or down), as a circle (daily, weekly, seasonally , and so on), and as irregular fluctuations (peaks or valleys) over time.
4. Simulation methods are a combination of causal and timeseries methods will imitate the behavior of consumers under different circumstances.
With the product of Cadbury’s Roses boxed chocolate, Purchasers of Costco’s confectionery Dept will forecast the consumption of this product base on 4th methods – simulation – that will be described in detail as below:
Simulation method comprises 2 methods causal and timeseries.
Base on causal method, chocolate’s demand is effected too much by:
1. Events: Christmas, Mother Day,
2. Price: better than other distributors or retailers.
3. Reputation of products and distributors.
In the other hand, with timeseries method, chocolates consumption forecast is subjected to:
* 4. Last year sales...

...TIMESERIES ANALYSIS
Chapter Three
Univariate TimeSeries Models
Chapter Three
Univariate timeseries models c WISE
1
3.1
Preliminaries
We denote the univariate timeseries of interest as yt.
• yt is observed for t = 1, 2, . . . , T ;
• y0, y−1, . . . , y1−p are available;
• Ωt−1 the history or information set at time t − 1.
Call such a sequence of random variables a timeseries.
Chapter Three
Univariate timeseries models c WISE
2
Martingales
Let {yt} denote a sequence of random variables and let It =
{yt, yt−1, . . .} denote a set of conditioning information or information
set based on the past history of yt. The sequence {yt, It} is called a
martingale if
• It−1 ⊂ It (It is a ﬁltration)
• E [|yt|] < ∞
• E [yt|It−1] = yt−1 (martingale property)
Chapter Three
Univariate timeseries models c WISE
3
Random walk model
The most common example of a martingale is the random walk model
yt = yt−1 + εt,
εt ∼ W N (0, σ 2)
where y0 is a ﬁxed initial value.
Letting It = {yt, . . . , y0} implies E [yt|It−1] = yt−1 since E [εt|It−1] = 0.
Chapter Three
Univariate timeseries models c WISE
4
Law of Iterated Expectations
Deﬁnition 1. In general, for information sets It and...