Time series data is data is collected for a single entity over time. This is fundamentally different from cross-section data which is data on multiple entities at the same point in time.

This is what econometricians call a dynamic causal effect. Let us go back to the application to cigarette consumption of Chapter 12 where we were interested in estimating the effect on cigarette demand of a price increase caused by a raise of the general sales tax.

One might use time series data to assess the causal effect of a tax increase on smoking both initially and in subsequent periods. Another application of time series data is forecasting. The remainder of Chapters in the book deals with the econometric techniques for the analysis of time series data and applications to forecasting and estimation of dynamic causal effects. This section covers the basic concepts presented in Chapter 14 of the book, explains how to visualize time series data and demonstrates how to estimate simple autoregressive models, where the regressors are past values of the dependent variable or other variables.

In this context we also discuss the concept of stationarity, an important property which has far-reaching consequences. Most empirical applications in this chapter are concerned with forecasting and use data on U. The following packages and their dependencies are needed for reproduction of the code chunks presented throughout this chapter:. Hlavac, M. Hyndman, R. Kleiber, C. Pfaff, B.

Kako prepoznati osobu koja baca urokeRyan, J. Wickham, H. Zeileis, A. Preface 1 Introduction 1. Computation of Heteroskedasticity-Robust Standard Errors 5.In statisticsPoisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables.

Poisson regression assumes the response variable Y has a Poisson distributionand assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters. A Poisson regression model is sometimes known as a log-linear modelespecially when used to model contingency tables. Negative binomial regression is a popular generalization of Poisson regression because it loosens the highly restrictive assumption that the variance is equal to the mean made by the Poisson model.

The traditional negative binomial regression model, commonly known as NB2, is based on the Poisson-gamma mixture distribution. This model is popular because it models the Poisson heterogeneity with a gamma distribution. Poisson regression models are generalized linear models with the logarithm as the canonical link functionand the Poisson distribution function as the assumed probability distribution of the response. Sometimes this is written more compactly as.

The maximum-likelihood estimates lack a closed-form expression and must be found by numerical methods. The probability surface for maximum-likelihood Poisson regression is always concave, making Newton—Raphson or other gradient-based methods appropriate estimation techniques.

Note that the expression on the right hand side has not actually changed. A formula in this form is typically difficult to work with; instead, one uses the log-likelihood :.

Poisson regression may be appropriate when the dependent variable is a count, for instance of events such as the arrival of a telephone call at a call centre. Poisson regression may also be appropriate for rate data, where the rate is a count of events divided by some measure of that unit's exposure a particular unit of observation.

For example, biologists may count the number of tree species in a forest: events would be tree observations, exposure would be unit area, and rate would be the number of species per unit area. More generally, event rates can be calculated as events per unit time, which allows the observation window to vary for each unit. In Poisson regression this is handled as an offsetwhere the exposure variable enters on the right-hand side of the equation, but with a parameter estimate for log exposure constrained to 1.

Offset in the case of a GLM in R can be achieved using the offset function:. A characteristic of the Poisson distribution is that its mean is equal to its variance. In certain circumstances, it will be found that the observed variance is greater than the mean; this is known as overdispersion and indicates that the model is not appropriate.

A common reason is the omission of relevant explanatory variables, or dependent observations. Under some circumstances, the problem of overdispersion can be solved by using quasi-likelihood estimation or a negative binomial distribution instead.How to install R. This booklet itells you how to use the R statistical software to carry out some simple analyses that are common in analysing time series data.

Slide in camper wiring diagram hd quality listThis booklet assumes that the reader has some basic knowledge of time series analysis, and the principal focus of the booklet is not to explain time series analysis, but rather to explain how to carry out these analyses using R. The first thing that you will want to do to analyse your time series data will be to read it into R, and to plot the time series.

You can read data into R using the scan function, which assumes that your data for successive time points is in a simple text file with one column. Only the first few lines of the file have been shown. The first three lines contain some comment on the data, and we want to ignore this when we read the data into R. To read the file into R, ignoring the first three lines, we type:. To store the data in a time series object, we use the ts function in R.

Sometimes the time series data set that you have may have been collected at regular intervals that were less than one year, for example, monthly or quarterly. An example is a data set of the number of births per month in New York city, from January to December originally collected by Newton. We can read the data into R by typing:. Once you have read a time series into R, the next step is usually to make a plot of the time series data, which you can do with the plot.

For example, to plot the time series of the age of death of 42 successive kings of England, we type:. We can see from the time plot that this time series could probably be described using an additive model, since the random fluctuations in the data are roughly constant in size over time.

We can see from this time series that there seems to be seasonal variation in the number of births per month: there is a peak every summer, and a trough every winter. Again, it seems that this time series could probably be described using an additive model, as the seasonal fluctuations are roughly constant in size over time and do not seem to depend on the level of the time series, and the random fluctuations also seem to be roughly constant in size over time.

Similarly, to plot the time series of the monthly sales for the souvenir shop at a beach resort town in Queensland, Australia, we type:. In this case, it appears that an additive model is not appropriate for describing this time series, since the size of the seasonal fluctuations and random fluctuations seem to increase with the level of the time series. Thus, we may need to transform the time series in order to get a transformed time series that can be described using an additive model.

**Introduction To Time Series In R: Measuring Predictive Model Quality**

For example, we can transform the time series by calculating the natural log of the original data:. Here we can see that the size of the seasonal fluctuations and random fluctuations in the log-transformed time series seem to be roughly constant over time, and do not depend on the level of the time series.

Thus, the log-transformed time series can probably be described using an additive model. Decomposing a time series means separating it into its constituent components, which are usually a trend component and an irregular component, and if it is a seasonal time series, a seasonal component. A non-seasonal time series consists of a trend component and an irregular component.The maximum likelihood estimation MLE method, typically used for polytomous logistic regression, is prone to bias due to both misclassification in outcome and contamination in the design matrix.

Hence, robust estimators are needed. In this study, we propose such a method for nominal response data with continuous covariates.

A generalized method of weighted moments GMWM approach is developed for dealing with contaminated polytomous response data. In this approach, distances are calculated based on individual sample moments. And Huber weights are applied to those observations with large distances.

Mellow-type weights are also used to downplay leverage points. We describe theoretical properties of the proposed approach.

Simulations suggest that the GMWM performs very well in correcting contamination-caused biases. An empirical application of the GMWM estimator on data from a survey demonstrates its usefulness. In practice, however, the model building process can be highly influenced by peculiarities in the data.

The maximum likelihood estimation MLE method, typically used for the polytomous logistic regression model PLRMis prone to bias due to both misclassification in outcome and contamination in the design matrix Pregibon, ; Copas, But all these methods are difficult to adapt for continuous covariates. The GMM is particularly useful when the moment conditions are relatively easy to obtain. Under some regularity conditions, the GMM estimator is consistent Hansen, Such observations can bring up disastrous bias on standard parameter estimates if they are not properly accounted for, see HuberHampel et al.

### Time Series Analysis Using ARIMA Model In R

So we propose a modified estimation method based on an outlier robust variant of GMM. The method is different from the kernel-weighted GMM developed for linear time-series data by Kuersteiner in that this is a data-driven method for defining weights.

The new approach is evaluated using asymptotic theory, simulations, and an empirical example. The robust GMM estimator is motivated by the data from a study on hypertension in a sample of the Chinese population.

Untold secret quoraObserved variables included demographics, social-economic status, weight, height, blood pressure, and food consumption. Sodium intakes were calculated based on overall food consumption. Among those covariates, age, body mass index BMIand sodium intakes are all continuous. Based on blood pressure measurements, subjects were classified into 4 categories: Normal, Pre-hypertension, Stage 1 and Stage 2 hypertension.

## Nonparametric Regression and Local Regression

Table 1 lists the summary statistics of the sample. One of the research objectives is to examine the association between hypertension and risk factors in the population.

Metal detecting utah ghost townsEach comparison have a set of parameters for all covariates in the model. Therefore, the generalized logit model is not parsimonious when comparing with the proportional odds model. But the simultaneous estimation of all parameters is more efficient than separate models for each comparison. It is another option for ordinal response data, especially when a proportional odds model does not fit the data well.

Table 2 lists the output from the model estimated by MLE. It is obvious that, if MLE is used, the estimates is inconsistent for sodium intakes, particularly the negative coefficient of sodium intake for the odds between the Stage 2 hypertension and the Normal categories.

Diagram based farmall m hydraulic valve diagramThe inconsistency is more obvious when we plot the odds with respect to the sodium intake, the downward trend of the odds in Fig. Besides, Fig. The scatter plot Fig.

Criteria c d for the distance and c x for the leverage are demonstrated. The paper is set up as follows.There are different techniques that are considered to be forms of nonparametric regression. Kendall—Theil regression fits a linear model between one x variable and one y variable using a completely nonparametric approach.

Quantile regression is a very flexible approach that can find a linear relationship between a dependent variable and one or more independent variables.

Local regression fits a smooth curve to the dependent variable, and can accommodate multiple independent variables. Generalized additive models are a powerful and flexible approach.

The following commands will install these packages if they are not already installed:. Nonparametric correlation is discussed in the chapter Correlation and Linear Regression.

Dell fingerprint softwareData for the examples in this chapter are borrowed from the Correlation and Linear Regression chapter. In this hypothetical example, students were surveyed for their weight, daily caloric intake, daily sodium intake, and a score on an assessment of knowledge gain.

## Poisson regression

Kendall—Theil regression is a completely nonparametric approach to linear regression where there is one independent and one dependent variable.

It is robust to outliers in the dependent variable. It simply computes all the lines between each pair of points, and uses the median of the slopes of these lines.

This method is sometimes called Theil—Sen. A modified, and preferred, method is named after Siegel. The method yields a slope and intercept for the fit line, and a p -value for the slope can be determined as well.

### Introduction to Parametric Tests

Typically, no measure analogous to r-squared is reported. The mblm function in the mblm package uses the Siegel method by default. See library mblm ;? MAD is the median absolute deviation, a robust measure of variability. While traditional linear regression models the conditional mean of the dependent variable, quantile regression models the conditional median or other quantile.

Medians are most common, but for example, if the factors predicting the highest values of the dependent variable are to be investigated, a 95 th percentile could be used. Likewise, models for several quantiles, e. Quantile regression makes no assumptions about the distribution of the underlying data, and is robust to outliers in the dependent variable.

It does assume the dependent variable is continuous. However, there are functions for other types of dependent variables in the qtools package.The following commands will install these packages if they are not already installed:. They include t -test, analysis of variance, and linear regression.

This might include variables measured in science such as fish length, child height, crop yield weight, or pollutant concentration in water. One advantage of using parametric statistical tests is that your audience will likely be familiar with the techniques and interpretation of the results.

These tests are also often more flexible and more powerful than their nonparametric analogues. Their major drawback is that all parametric tests assume something about the distribution of the underlying data.

If these assumptions are violated, the resultant test statistics will not be valid, and the tests will not be as powerful as for cases when assumptions are met. A frequent error is to use common parametric models and tests with count data for the dependent variable. Instead, count data could be analyzed either by using tests for nominal data or by using regression methods appropriate for count data.

These include Poisson regression, negative binomial regression, and zero-inflated Poisson regression. See the Regression for Count Data chapter. It is sometimes permissible to use common parametric tests for count data or other discrete data. They can be used in cases where counts are used as a type of measurement of some property of subjects, provided that 1 the distribution of data or residuals from the analysis approximately meet test assumptions; and 2 there are few or no counts at or close to zero, or close to a maximum, if one exists.

Permissible examples might include test scores, age, or number of steps taken during the day. Technically, each of these measurements is bound by zero, and are discrete rather than continuous measurements. However, if other conditions are met, it is reasonable to handle them as if they were continuous measurement variables. This kind of count data will sometimes need to be transformed to meet the assumptions of parametric analysis.

Square root and logarithmic transformations are common. However, if there are many counts at or near zero, transformation is unlikely to help. It is usually not worth the effort to attempt to force count data to meet the assumptions of parametric analysis with transformations, since there are more appropriate methods available.Time series data are data points collected over a period of time as a sequence of time gap.

Time series data analysis means analyzing the available data to find out the pattern or trend in the data to predict some future values which will, in turn, help more effective and optimize business decisions.

Moreover, time series analysis can be classified as:. Auto Regressive AR terms refer to the lags of the differenced series, Moving Average MA terms refer to the lags of errors and I is the number of difference used to make the time series stationary.

The first step in time series data modeling using R is to convert the available data into time series data format. To do so we need to run the following command in R:. This is how the actual dataset looks like:. We can infer from the graph itself that the data points follows an overall upward trend with some outliers in terms of sudden lower values. Now we need to do some analysis to find out the exact non-stationary and seasonality in the data.

Before performing any EDA on the data, we need to understand the three components of a time series data:. The output will look like this:. Observing these 4 graphs closely, we can find out if the data satisfies all the assumptions of ARIMA modeling, mainly, stationarity and seasonality. For the sake of discussion here, we will remove the seasonal part of the data as well. The seasonal part can be removed from the analysis and added later, or it can be taken care of in the ARIMA model itself.

After removing non-stationarity:.

The autocorrelation function acf gives the autocorrelation at all possible lags. The autocorrelation at lag 0 is included by default which always takes the value 1 as it represents the correlation between the data and themselves. As we can infer from the graph above, the autocorrelation continues to decrease as the lag increases, confirming that there is no linear association between observations separated by larger lags.

To remove seasonality from the data, we subtract the seasonal component from the original series and then difference it to make it stationary. After removing seasonality and making the data stationary, it will look like:. Smoothing is usually done to help us better see patterns, trends in time series. Generally it smooths out the irregular roughness to see a clearer signal. For seasonal data, we might smooth out the seasonality so that we can identify the trend. Once the data is ready and satisfies all the assumptions of modeling, to determine the order of the model to be fitted to the data, we need three variables: p, d, and q which are non-negative integers that refer to the order of the autoregressive, integrated, and moving average parts of the model respectively.

To examine which p and q values will be appropriate we need to run acf and pacf function. The plots will look like:. Shape of acf to define values of p and q: Looking at the graphs and going through the table we can determine which type of the model to select and what will be the values of p, d and q. The default is conditional-sum-of-squares. This is a recursive process and we need to run this arima function with different p,d,q values to find out the most optimized and efficient model.

The output from fitarima includes the fitted coefficients and the standard error s. Observing the coefficients we can exclude the insignificant ones. We can use a function confint for this purpose.

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