What Is an Autoregressive Integrated Moving Average (ARIMA)?

An autoregressive integrated moving average, or ARIMA, is a statistical analysis model that uses time series data to either better understand the data set or to predict future trends. 

A statistical model is autoregressive if it predicts future values based on past values. For example, an ARIMA model might seek to predict a stock’s future prices based on its past performance or forecast a company’s earnings based on past periods.

Understanding Autoregressive Integrated Moving Average (ARIMA)

An autoregressive integrated moving average model is a form of regression analysis that gauges the strength of one dependent variable relative to other changing variables. The model’s goal is to predict future securities or financial market moves by examining the differences between values in the series instead of through actual values.

An ARIMA model can be understood by outlining each of its components as follows:

• Autoregression (AR): refers to a model that shows a changing variable that regresses on its own lagged, or prior, values.
• Integrated (I): represents the differencing of raw observations to allow the time series to become stationary (i.e., data values are replaced by the difference between the data values and the previous values).
• Moving average (MA):  incorporates the dependency between an observation and a residual error from a moving average model applied to lagged observations.

ARIMA Parameters

Each component in ARIMA functions as a parameter with a standard notation. For ARIMA models, a standard notation would be ARIMA with p, d, and q, where integer values substitute for the parameters to indicate the type of ARIMA model used. The parameters can be defined as:

• p: the number of lag observations in the model, also known as the lag order.
• d: the number of times the raw observations are differenced; also known as the degree of differencing.
• q: the size of the moving average window, also known as the order of the moving average.

For example, a linear regression model includes the number and type of terms. A value of zero (0), which can be used as a parameter, would mean that particular component should not be used in the model. This way, the ARIMA model can be constructed to perform the function of an ARMA model, or even simple AR, I, or MA models.

ARIMA and Stationary Data

In an autoregressive integrated moving average model, the data are differenced in order to make it stationary. A model that shows stationarity is one that shows there is constancy to the data over time. Most economic and market data show trends, so the purpose of differencing is to remove any trends or seasonal structures. 

Seasonality, or when data show regular and predictable patterns that repeat over a calendar year, could negatively affect the regression model. If a trend appears and stationarity is not evident, many of the computations throughout the process cannot be made and produce the intended results.

How to Build an ARIMA Model

To begin building an ARIMA model for an investment, you download as much of the price data as you can. Once you’ve identified the trends for the data, you identify the lowest order of differencing (d) by observing the autocorrelations. If the lag-1 autocorrelation is zero or negative, the series is already differenced. You may need to difference the series more if the lag-1 is higher than zero.

Next, determine the order of regression (p) and order of moving average (q) by comparing autocorrelations and partial autocorrelations. Once you have the information you need, you can choose the model you’ll use.

Pros and Cons of ARIMA

ARIMA models have strong points and are good at forecasting based on past circumstances, but there are more reasons to be cautious when using ARIMA. In stark contrast to investing disclaimers that state “past performance is not an indicator of future performance…,” ARIMA models assume that past values have some residual effect on current or future values and use data from the past to forecast future events.

The following table lists other ARIMA traits that demonstrate good and bad characteristics.

The Bottom Line

The ARIMA model is used as a forecasting tool to predict how something will act in the future based on past performance. It is used in technical analysis to predict an asset’s future performance.

ARIMA modeling is generally inadequate for long-term forecastings, such as more than six months ahead, because it uses past data and parameters that are influenced by human thinking. For this reason, it is best used with other technical analysis tools to get a clearer picture of an asset’s performance.

What Is an Auditor’s Report?

An auditor’s report is a written letter from the auditor containing their opinion on whether a company’s financial statements comply with generally accepted accounting principles (GAAP) and are free from material misstatement.

The independent and external audit report is typically published with the company’s annual report. The auditor’s report is important because banks and creditors require an audit of a company’s financial statements before lending to them.

How an Auditor’s Report Works

An auditor’s report is a written letter attached to a company’s financial statements that expresses its opinion on a company’s compliance with standard accounting practices. The auditor’s report is required to be filed with a public company’s financial statements when reporting earnings to the Securities and Exchange Commission (SEC).

However, an auditor’s report is not an evaluation of whether a company is a good investment. Also, the audit report is not an analysis of the company’s earnings performance for the period. Instead, the report is merely a measure of the reliability of the financial statements.

The Components of an Auditor’s Report

The auditor’s letter follows a standard format, as established by generally accepted auditing standards (GAAS). A report usually consists of three paragraphs.

• The first paragraph states the responsibilities of the auditor and directors.
• The second paragraph contains the scope, stating that a set of standard accounting practices was the guide.
• The third paragraph contains the auditor’s opinion.

An additional paragraph may inform the investor of the results of a separate audit on another function of the entity. The investor will key in on the third paragraph, where the opinion is stated.

The type of report issued will be dependent on the findings by the auditor. Below are the most common types of reports issued for companies.

Clean or Unqualified Report

A clean report means that the company’s financial records are free from material misstatement and conform to the guidelines set by GAAP. A majority of audits end in unqualified, or clean, opinions.

Qualified Opinion

A qualified opinion may be issued in one of two situations: first, if the financial statements contain material misstatements that are not pervasive; or second, if the auditor is unable to obtain sufficient appropriate audit evidence on which to base an opinion, but the possible effects of any material misstatements are not pervasive. For example, a mistake might have been made in calculating operating expenses or profit. Auditors typically state the specific reasons and areas where the issues are present so that the company can fix them.

Adverse Opinion

An adverse opinion means that the auditor has obtained sufficient audit evidence and concludes that misstatements in the financial statements are both material and pervasive. An adverse opinion is the worst possible outcome for a company and can have a lasting impact and legal ramifications if not corrected.

Disclaimer of Opinion

A disclaimer of opinion means that, for some reason, the auditor is unable to obtain sufficient audit evidence on which to base the opinion, and the possible effects on the financial statements of undetected misstatements, if any, could be both material and pervasive. Examples can include when an auditor can’t be impartial or wasn’t allowed access to certain financial information.

Example of an Auditor’s Report

Excerpts from the audit report by Deloitte & Touche LLP for Starbucks Corporation, dated Nov. 15, 2019, follow.

Paragraph 1: Opinion on the Financial Statements

“We have audited the accompanying consolidated balance sheets of Starbucks Corporation and subsidiaries (the ‘Company’) as of September 29, 2019, and September 30, 2018, the related consolidated statements of earnings, comprehensive income, equity, and cash flows, for each of the three years in the period ended September 29, 2019, and the related notes (collectively referred to as the ‘financial statements’).

In our opinion, the financial statements present fairly, in all material respects, the financial position of the Company as of September 29, 2019, and September 30, 2018, and the results of its operations and its cash flows for each of the three years in the period ended September 29, 2019, in conformity with accounting principles generally accepted in the United States of America.”

Paragraph 2: Basis for Opinion

“We conducted our audits in accordance with the standards of the Public Company Accounting Oversight Board (PCAOB). Those standards require that we plan and perform the audit to obtain reasonable assurance about whether the financial statements are free of material misstatement, whether due to error or fraud. Our audits included performing procedures to assess the risks of material misstatement of the financial statements, whether due to error or fraud, and performing procedures that respond to those risks.

Such procedures included examining, on a test basis, evidence regarding the amounts and disclosures in the financial statements. Our audits also included evaluating the accounting principles used and significant estimates made by management, as well as evaluating the overall presentation of the financial statements. We believe that our audits provide a reasonable basis for our opinion.”

What Is an Autoregressive Model?

A statistical model is autoregressive if it predicts future values based on past values. For example, an autoregressive model might seek to predict a stock’s future prices based on its past performance.

Understanding Autoregressive Models

Autoregressive models operate under the premise that past values have an effect on current values, which makes the statistical technique popular for analyzing nature, economics, and other processes that vary over time. Multiple regression models forecast a variable using a linear combination of predictors, whereas autoregressive models use a combination of past values of the variable.

An AR(1) autoregressive process is one in which the current value is based on the immediately preceding value, while an AR(2) process is one in which the current value is based on the previous two values. An AR(0) process is used for white noise and has no dependence between the terms. In addition to these variations, there are also many different ways to calculate the coefficients used in these calculations, such as the least squares method.

These concepts and techniques are used by technical analysts to forecast security prices. However, since autoregressive models base their predictions only on past information, they implicitly assume that the fundamental forces that influenced the past prices will not change over time. This can lead to surprising and inaccurate predictions if the underlying forces in question are in fact changing, such as if an industry is undergoing rapid and unprecedented technological transformation.

Nevertheless, traders continue to refine the use of autoregressive models for forecasting purposes. A great example is the Autoregressive Integrated Moving Average (ARIMA), a sophisticated autoregressive model that can take into account trends, cycles, seasonality, errors, and other non-static types of data when making forecasts.

Example of an Autoregressive Model

Autoregressive models are based on the assumption that past values have an effect on current values. For example, an investor using an autoregressive model to forecast stock prices would need to assume that new buyers and sellers of that stock are influenced by recent market transactions when deciding how much to offer or accept for the security.

Although this assumption will hold under most circumstances, this is not always the case. For example, in the years prior to the 2008 Financial Crisis, most investors were not aware of the risks posed by the large portfolios of mortgage-backed securities held by many financial firms. During those times, an investor using an autoregressive model to predict the performance of U.S. financial stocks would have had good reason to predict an ongoing trend of stable or rising stock prices in that sector. 

However, once it became public knowledge that many financial institutions were at risk of imminent collapse, investors suddenly became less concerned with these stocks’ recent prices and far more concerned with their underlying risk exposure. Therefore, the market rapidly revalued financial stocks to a much lower level, a move which would have utterly confounded an autoregressive model.

It is important to note that, in an autoregressive model, a one-time shock will affect the values of the calculated variables infinitely into the future. Therefore, the legacy of the financial crisis lives on in today’s autoregressive models.

Investopedia does not provide tax, investment, or financial services and advice. The information is presented without consideration of the investment objectives, risk tolerance, or financial circumstances of any specific investor and might not be suitable for all investors. Investing involves risk, including the possible loss of principal.