Non– Stationary Model
Introduction
Corporations and financial institutions as well as researchers and individual investors often use financial time series data such as exchange rates, asset prices, inflation, GDP and other macroeconomic indicator in the analysis of stock market, economic forecasts or studies of the data itself (Kitagawa, G., & Akaike, H, 1978). To apply data refining data is the key as it enables one to isolate the data points that are relevant to your stock reports. The data points have covariance, variances and mean that changes with time that is the data are non – stationary. Non – stationary behavior can be cycles, random walks, trends or the combination of the three. Non – stationary data are unpredictable and cannot be forecasted or modeled. The result achieved by using non – stationary time series may indicate a relationship between two variables where one does not exist so they may be spurious. In order to receive the reliable result and also consistent the non – stationary data need to be transformed into stationary data. In contrast to the non – stationary process that has a variable mean and a variance that does not remain near or returns to a long-run mean over time, the stationary process reverts around a constant variance and has a constant long term means independent of time.
A stationary time series has a constant variance, a constant mean and the variance is independent of time. Without stationery is important for standard econometric theory and is also crucial for one to obtain consistent estimators. Plotting the series against time is the quickest way of telling if a process is stationary. Chances are that the series is stationary if the graph crosses the mean of the sample many times. If the graph does not cross the mean is an indication of persistent trends away from the mean of the series. A variable whose it mean grows around a fixed trend is a trend stationary variable. This enables a classical way of describing an economical time series which grows at a constant rate. A trend series generally tends to revolve around a steady, up warding sloping curve that does not have big swings away from that curve. De trending the series give a stationary process.
The following types of non – stationary processes are the possible candidates:
The examples of non – stationary processes are random walk without drift, random walk with drift and deterministic trends.
Random walk without drift or pure random walk (Y t = Yt-1 + ε t )
Random walk predicts that the value at a time “t” will be equal to the previous period value plus a stochastic component that is white noise. Stochastic component is identically and independently distributed with mean “0” and variance “σ²”. The random walk can also be named as processor unit with a stochastic trend or a process integrated of some order. The process can either move in a negative or positive direction from the mean hence it is a non-mean reverting process. The random walk goes to the infinity as the time goes to the infinity and also evolves over time; this makes a random walk cannot be predicted. In a purely random process, every element is independent of every other element and each element has an identical distribution.
Random walk with drift (Y t = α + Yt-1 + ε t )
The random walk with a drift predicts that the value at a time “t” will equal the last previous period’s value plus a drift or a constant and a white noise term. Random walk with a drift does not revert to a long-run mean and has variance dependent on time. Stochastic process in this case are defined in terms of its behavior at time t where t is an integer value. Since t is a quantity that can be mapped to the integer then there is no problem but essential difference occurs if t is a continuous variable. One considers the value of the time series at a given time to be continuous random variable. Many times series mostly those from volcanology take other kinds of values, for example, the kinds of discrete events or count values such as those recording number of eruption.
Deterministic trend (Y t = α + βt + ε t )
Deterministic trend predicts that the value at a time “t” will equal the last previous period’s value plus a drift or a constant and a white noise same as random walk with a drift. The value at time “t” in the case of a deterministic trend it is regressed on a time trend (βt) while in the case of a random walk is regressed on a time the last period’s value (Yt-1). The deterministic trend is a non – stationary process that has a mean that grows around a fixed trend, which is independent of time and constant.
Random walk with drift and deterministic trend (Y t = α + Yt-1 + βt + ε t )
Random walk with drift and deterministic trend is a non – stationary process that combines a random walk with a deterministic trend and a drift component. The process specifies the value a time “t” by the drift, a stochastic component, a trend, and a last period value.
Trend and difference stationary
A random walk without or with a drift can be transformed to a stationary process by differencing (subtracting Yt-1 from Y t, taking the difference Y t – Yt-1) correspondingly to Y t – Yt-1 = ε t or Y t – Yt-1= α + ε t and then the process becomes difference-stationary. The process loses one observation each time the difference is taken and is the disadvantage of differencing stationary. A non – stationary process with a deterministic trend becomes stationary only after removing de trending or the trend. For example, deterministic trend is transformed into stationary process by subtracting the trend (Cowpertwait, P. S., & Metcalfe, A. V, 2009). When one transfers non – stationary to stationary through de trending no observation is lost. In the case of a random walk with a deterministic trend and a drift, de trending may remove both the drift and deterministic trend but the variance will go to the infinity. Due to this, the differencing must also be applied to remove the stochastic trend.
Table 1 Differencing as shown in table 1 there is difference between the curve of a random walk with drift and the difference stationary. The advantage of the difference is that the method removes the trend in the variance, stochastic trend and also differencing is simple to use. The disadvantage is that differencing process losses one observation each time the difference is taken.
Table 2 de trending or removing the trend
After removing the trend, non – stationary process with deterministic trend becomes stationary the process is also known as de trending. The trend βt is subtracted from the non- stationary process α + βt + ε t to be transformed into a stationary process α + ε t. When removing trend to transform a non – stationary process to a stationary one no observation are lost. Removing trend can remove the drift and the deterministic trend, whereas the variance will continue to the infinity. For one to remove the stochastic trend the differencing must also be applied.
In financial model, using non – stationary time series data produces spurious and unreliable result and leads to poor forecasting and understanding. The problem to be solved is to transform the time series data so that it becomes stationary. Differencing is used to transform non – stationary process with a drift or a random walk without or with a drift to stationary process. If the time data being analyzed contain a deterministic trend, the spurious result is avoided by de trending. The non – stationary series may combine a deterministic and stochastic trend at the same time and to avoid obtaining misleading result both the de trending and differencing are applies. De trending removes the deterministic trend and the differencing removes the trend in the variance.
One should always check whether two variables are integrated the same way to obtain meaningful result. Stationary stochastic process is the one where given t1 , . . . ., t l the joint statistical distribution of Xt1, . . . . . . . , X tl is the same as the joint statistical distribution of Xt1+r, . . . . ., X tl + r for all l and r. this means that all moments of all degrees that is variances, expectation, third-order and higher of the process anywhere are the same. This also means that the joint distribution of ( X t + r , X) is the same as the ( X t, X s ) and hence cannot depend on s or t but only on the distance between s and t that is s – t. This stationarity is too strict for everyday life a weaker of weaker stationarity or second order is usually used. Second order means that variance of stochastic process and mean do not depend on t.
Global non- stationarity
The random walk is example of a global non – stationarity. The parameter and process of global non-stationarity process are fixed some time ago generally infinity and then the process itself evolves whereas the rules of evolution does not change.
AR, MA and ARMA models
This segment considers some primary probability models that are extensively used for modeling a given time series.
Moving Average models.
Probably the next simplest model is that constructed by the simplest linear combinations of dawdled elements of a totally purely random process, {ǫ t} with Eǫt = 0. A moving mean process, {Xt}, of the order q is then defined as .
………………………………………………………………… equation 11.5
And the abbreviated notation is noted as MA (q). Normally, with a newly redefined procedure it is of interest to note that all its statistical properties. For an MA (q) process, the average is simple to compute, since all the expectation of a given sum is the sum of all the expectations):
……. equation 11.6
This is simply because E(ǫ r) = 0 for any and all values of r. A totally similar argument can be applied for the divergent calculation
……………………………………………………….. equation 11.7
Since var [(ǫ r ) = σ 2] for all values of t
The sample distribution of the autocorrelation can be calculated as [r (τ ) = c (τ ) / c (0)]. If, when one calculates the sample auto covariance, it chops off at a particular lag q, for example it is effectively and adequately zero for lags of [ q + 1 ] or higher, then one can contend the MA(q) model in above equation as the implicit in probability model.
The only most quantity that we should concern ourselves with is only the auto covariance. The cheapest way of seeing what the auto covariance is to take into assumption that the procedures is stationary and multiply in both sides of (figure 11.17) by Xt−τ and take expectations:
This equates to Qt is independent of [X t− τ] and hence [E(ǫ t X t− τ ) = E ǫ t E X t− τ = 0] since the totally purely random procedure here has zero average. Thus (equation11.35) turns into
The formula (equation11.3) is a simplest example of a well-known Yule-Walker equation, more complex versions can be used to obtain a formulae for the auto covariance of more universal AR (p) procedures.
There are several checks and try outs that one can easily make but the absolute comparison of the try outs auto covariance with source values, such as the model of auto covariance given in equation below, is a major first step in model identification.
Also, at this point one can question what one means by the term “effectively zero”. The try out auto covariance is an empirically and statistically computed from the random sample distribution at hand. If more informational data in the time series were to be collected, or another try out stretch used then the sample auto covariance would be greatly different (although for extremely long samples and stationary time series the probability of a huge difference should be much smaller). Hence, try out autocovariance (autocorrelations) are significant random quantities and hence is “effectively zero” render into a continuous statistical hypothesis tests on whether the true auto correlation is equivalent to zero or not.
Finally, while we are still on the subject of sample autocovariance we notice that at the extremes points of the range of τ
……………..equation 11.4
Which also a times happens to be this sample variance and
……… equation 11.5
The horizontally oriented dashed lines in the bottom plot of the above figure [Figure 11.1] are approximately 95% significance bands. Any function which is outside any of these bands (like the ones shown in the figure at specified lags 1 and 2) are viewed as very important, or at least worth further rumination. It is also interesting to realize that although [ρ (τ ) = 0 ] for [ τ > 2] the sample autocorrelations for all lags 10 to 14 inclusive all look quite huge: This is merely an demonstration of the point out that the sample ac f function is a totally random quantity and some extensive care and a large enough experience is required to prevent excessive reading into them.
Autoregressive models.
The other primary development from a totally pure random procedure is the autoregressive model, which, as its name suggests, models a procedure where future values somehow will depend on the recent epoch. Moreover, {X t} follows an AR(p) model which is qualified by
Where {ǫt} is a totally pure random procedure. For the sake of this discussion we shall adopt time-honored history and firstly put into consideration the most simplest AR model: AR(1). Here we simplify notation and define {Xt} by
Since we the ǫt are different and independent
Then with respect to other equations not necessarily indicated above allude that final quantity that we need to worry ourselves with is the only auto-covariance (Schneider, R., & Weil, W, 2008). The simplest way of seeing what the auto covariance actually is, is to assume that the whole process is totally stationary and multiply both sides of (equation 11.17) by [ X t – τ] and take expectations:
We have simulated in total two AR (1) processes each of 200 different observations. The first [ar1] processes is a realization insinuating that [α = 0.9] and the second,[ar 1neg] is an actualization where[ α = −0.9]. Plots of each subsequent actualization and their autocorrelation coefficients in the functions appear. The total exponential decay nature of the auto correlation of [ar1pos] is clear and one can easily note that [ρ (1) = α = 0.9] from the plot section. For [ar1neg] similar considerations do apply except that [ρ(τ )] has an extra [(−1) τ] factor which results in the oscillation
ARMA Models.
It is by a fact that both AR and MA models expresses different behaviors of stochastic interdependence. AR procedures encapsulate a Markov-like quality such that the future events depend on the past, whereas MA processes bring into combination of both elements of randomness from the past events using a non-static moving window. An evident step is to combine all both types of behavior into one ARMA (p, q) model which is found by purely simple concatenations. The procedure [X t] is ARMA(p, q) if
Fitting ARMA models.
By now, we have just identified the probability models implicit in ARMA processes and some other mathematical quantities that are related to them. Of course, once one has enough real data one would like to correctly know the answers to questions such as “Are all these models suitable for my data?”, “Can I accommodate all these models in my data?” “How do I fit these models?”, “Do the first model fit to the data well?”. These kind of questions lead directly to the Box-Jenkins procedure, see Box et al. (1994). Part of this procedure involves analyzing the sample autocorrelation coefficient functions (and are related the function is usually known as partial autocorrelation function) to settle on the order of any AR or MA terms. However, there are several other prospectors such as getting rid of deterministic courses, removing occupants, or/and checking residues. Actually, a full intervention is way beyond the length restrains of this article so we prefer referring the any interested reader to The Chatfield (2003) in the first and preceding instance.
Conclusion
Differencing transforms non – stationary process with a drift or a random walk without or with a drift to stationary process. If the time data being analyzed contain a deterministic trend, the spurious result is avoided by de trending. The non – stationary series may combine a deterministic and stochastic trend at the same time and to avoid obtaining misleading result both the de trending and differencing are applies. De trending removes the deterministic trend and the differencing removes the trend in the variance.
References
Kitagawa, G., & Akaike, H. (1978). A procedure for the modeling of non-stationary time series. Annals of the Institute of Statistical Mathematics, 30(1), 351-363.
Cowpertwait, P. S., & Metcalfe, A. V. (2009). Non-stationary Models. InIntroductory Time Series with R (pp. 137-157). Springer New York.
Schneider, R., & Weil, W. (2008). Non-stationary Models. Stochastic and Integral Geometry, 521-556.
Get Professional Assignment Help Cheaply
Are you busy and do not have time to handle your assignment? Are you scared that your paper will not make the grade? Do you have responsibilities that may hinder you from turning in your assignment on time? Are you tired and can barely handle your assignment? Are your grades inconsistent?
Whichever your reason is, it is valid! You can get professional academic help from our service at affordable rates. We have a team of professional academic writers who can handle all your assignments.
Why Choose Our Academic Writing Service?
- Plagiarism free papers
- Timely delivery
- Any deadline
- Skilled, Experienced Native English Writers
- Subject-relevant academic writer
- Adherence to paper instructions
- Ability to tackle bulk assignments
- Reasonable prices
- 24/7 Customer Support
- Get superb grades consistently
Online Academic Help With Different Subjects
Literature
Students barely have time to read. We got you! Have your literature essay or book review written without having the hassle of reading the book. You can get your literature paper custom-written for you by our literature specialists.
Finance
Do you struggle with finance? No need to torture yourself if finance is not your cup of tea. You can order your finance paper from our academic writing service and get 100% original work from competent finance experts.
Computer science
Computer science is a tough subject. Fortunately, our computer science experts are up to the match. No need to stress and have sleepless nights. Our academic writers will tackle all your computer science assignments and deliver them on time. Let us handle all your python, java, ruby, JavaScript, php , C+ assignments!
Psychology
While psychology may be an interesting subject, you may lack sufficient time to handle your assignments. Don’t despair; by using our academic writing service, you can be assured of perfect grades. Moreover, your grades will be consistent.
Engineering
Engineering is quite a demanding subject. Students face a lot of pressure and barely have enough time to do what they love to do. Our academic writing service got you covered! Our engineering specialists follow the paper instructions and ensure timely delivery of the paper.
Nursing
In the nursing course, you may have difficulties with literature reviews, annotated bibliographies, critical essays, and other assignments. Our nursing assignment writers will offer you professional nursing paper help at low prices.
Sociology
Truth be told, sociology papers can be quite exhausting. Our academic writing service relieves you of fatigue, pressure, and stress. You can relax and have peace of mind as our academic writers handle your sociology assignment.
Business
We take pride in having some of the best business writers in the industry. Our business writers have a lot of experience in the field. They are reliable, and you can be assured of a high-grade paper. They are able to handle business papers of any subject, length, deadline, and difficulty!
Statistics
We boast of having some of the most experienced statistics experts in the industry. Our statistics experts have diverse skills, expertise, and knowledge to handle any kind of assignment. They have access to all kinds of software to get your assignment done.
Law
Writing a law essay may prove to be an insurmountable obstacle, especially when you need to know the peculiarities of the legislative framework. Take advantage of our top-notch law specialists and get superb grades and 100% satisfaction.
What discipline/subjects do you deal in?
We have highlighted some of the most popular subjects we handle above. Those are just a tip of the iceberg. We deal in all academic disciplines since our writers are as diverse. They have been drawn from across all disciplines, and orders are assigned to those writers believed to be the best in the field. In a nutshell, there is no task we cannot handle; all you need to do is place your order with us. As long as your instructions are clear, just trust we shall deliver irrespective of the discipline.
Are your writers competent enough to handle my paper?
Our essay writers are graduates with bachelor's, masters, Ph.D., and doctorate degrees in various subjects. The minimum requirement to be an essay writer with our essay writing service is to have a college degree. All our academic writers have a minimum of two years of academic writing. We have a stringent recruitment process to ensure that we get only the most competent essay writers in the industry. We also ensure that the writers are handsomely compensated for their value. The majority of our writers are native English speakers. As such, the fluency of language and grammar is impeccable.
What if I don’t like the paper?
There is a very low likelihood that you won’t like the paper.
Reasons being:
- When assigning your order, we match the paper’s discipline with the writer’s field/specialization. Since all our writers are graduates, we match the paper’s subject with the field the writer studied. For instance, if it’s a nursing paper, only a nursing graduate and writer will handle it. Furthermore, all our writers have academic writing experience and top-notch research skills.
- We have a quality assurance that reviews the paper before it gets to you. As such, we ensure that you get a paper that meets the required standard and will most definitely make the grade.
In the event that you don’t like your paper:
- The writer will revise the paper up to your pleasing. You have unlimited revisions. You simply need to highlight what specifically you don’t like about the paper, and the writer will make the amendments. The paper will be revised until you are satisfied. Revisions are free of charge
- We will have a different writer write the paper from scratch.
- Last resort, if the above does not work, we will refund your money.
Will the professor find out I didn’t write the paper myself?
Not at all. All papers are written from scratch. There is no way your tutor or instructor will realize that you did not write the paper yourself. In fact, we recommend using our assignment help services for consistent results.
What if the paper is plagiarized?
We check all papers for plagiarism before we submit them. We use powerful plagiarism checking software such as SafeAssign, LopesWrite, and Turnitin. We also upload the plagiarism report so that you can review it. We understand that plagiarism is academic suicide. We would not take the risk of submitting plagiarized work and jeopardize your academic journey. Furthermore, we do not sell or use prewritten papers, and each paper is written from scratch.
When will I get my paper?
You determine when you get the paper by setting the deadline when placing the order. All papers are delivered within the deadline. We are well aware that we operate in a time-sensitive industry. As such, we have laid out strategies to ensure that the client receives the paper on time and they never miss the deadline. We understand that papers that are submitted late have some points deducted. We do not want you to miss any points due to late submission. We work on beating deadlines by huge margins in order to ensure that you have ample time to review the paper before you submit it.
Will anyone find out that I used your services?
We have a privacy and confidentiality policy that guides our work. We NEVER share any customer information with third parties. Noone will ever know that you used our assignment help services. It’s only between you and us. We are bound by our policies to protect the customer’s identity and information. All your information, such as your names, phone number, email, order information, and so on, are protected. We have robust security systems that ensure that your data is protected. Hacking our systems is close to impossible, and it has never happened.
How our Assignment Help Service Works
1. Place an order
You fill all the paper instructions in the order form. Make sure you include all the helpful materials so that our academic writers can deliver the perfect paper. It will also help to eliminate unnecessary revisions.
2. Pay for the order
Proceed to pay for the paper so that it can be assigned to one of our expert academic writers. The paper subject is matched with the writer’s area of specialization.
3. Track the progress
You communicate with the writer and know about the progress of the paper. The client can ask the writer for drafts of the paper. The client can upload extra material and include additional instructions from the lecturer. Receive a paper.
4. Download the paper
The paper is sent to your email and uploaded to your personal account. You also get a plagiarism report attached to your paper.
PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET A PERFECT SCORE!!!
