Random Walk Model (2016)

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The random walk model is widely used in the area of finance. The stock prices or exchange rates (Asset prices) follow a random walk. A common and serious departure from random behavior is called a random walk (non-stationary) since today’s stock price is equal to yesterday’s stock price plus a random shock.

Types of Random Walk Model

There are two types of random walks

  1. Random walk without drift (no constant or intercept)
  2. Random walk with drift (with a constant term)

Definition

A time series is said to follow a random walk if the first differences (difference from one observation to the next observation) are random.

Note that in a random walk model, the time series itself is not random, however, the first differences in time series are random (the differences change from one period to the next).

A random walk model for a time series $X_t$ can be written as

[X_t=X_{t-1}+e_t, , ,]

where $X_t$ is the value in time period $t$, $X_{t-1}$ is the value in time period $t-1$ plus a random shock $e_t$ (value of error term in time period $t$).

Since the random walk is defined in terms of first differences, therefore, it is easier to see the model as

[X_t-X_{t-1}=e_t, , ,]

where the original time series is changed to a first difference time series, that is the time series is transformed.

The transformed time series:

  • Forecast the future trends to aid in decision-making
  • If the time series follows a random walk, the original series offers little or no insights
  • May need to analyze the first differenced time series

Real World Example

Consider a real-world example of the daily US-dollar-to-Euro exchange rate. A plot of the entire history (of daily US-dollar-to-Euro exchange rate) from January 1, 1999, to December 5, 2014, looks like

Random Walk model

The historical pattern from the above plot looks quite interesting, with many peaks and valleys. The plot of the daily changes (first difference) would look like

Random Walk Model first difference

The volatility (variance) has not been constant over time, but the day-to-day changes are almost completely random.

Key Characteristics of a Random Walk

  • No Pattern: The path taken by a random walk is unpredictable.
  • Independence: Each step is independent of the previous one.
  • Probability distribution: The size and direction of each step can be defined by a probability distribution.

Applications of Random Walk Models

Beyond finance, random walk models have applications in:

  • Physics: Brownian motion and diffusion processes
  • Biology: Population dynamics and genetic drift
  • Computer science: Algorithms and simulations

Remember that, random walk patterns are also widely found elsewhere in nature, for example, in the phenomenon of Brownian Motion that was first explained by Einstein.

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Creating Formulas in MS Excel and Changing Data

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Changing the data

Before Creating formulas in MS Excel, you need numeric data in different columns or rows of the Excel sheet. Suppose you want to enter a few numbers in a column. Before entering these numbers, you should confirm the cell reference where you need to enter the data. Let’s start by entering numbers in Microsoft Excel‘s cells $A1$ and $A2$. For this purpose follow the steps given below

  1. Click on the cell $A1$
  2. Type 3 from the keyboard
  3. Press the ENTER or DOWN ARROW key on the keyboard. You will be in Cell A2
  4. Now type say 2 from the keyboard and press the ENTER key

Suppose you want to add these numbers in Cell $C1$. You need to write a formula in cell $C1$. After writing the correct formula the content of Cell $C1$ will immediately change to adding two numbers typed in $A1$ and $A2$ and used in $C1$ as formula content.

excel-data-and-formula

Creating Formulas in MS Excel

In this section, we will learn about Creating Formulas in MS Excel. In Excel, each formula begins with an equal sign ($=$), see the picture below

Creating Formulas in MS Excel

Therefore, when creating formulas in Excel, ALWAYS start by typing the equal sign. The equal sign is typed in the Cell where you want the answer to appear. Like the image above, follow these steps

  1. Click on cell $C1$ with ARROW keys from the keyboard or with the mouse pointer.
  2. Type the equal sign in cell $C1$.
excel-data-and-formula

After typing the equal sign in step 2, you have two choices for adding cell references to the spreadsheet formula. Note that cell reference is the name of the cell you want to use in the formula. $A1$ and $A2$ are cell references of numbers 3 and 2, respectively.

  1. You can type these references in or,
  2. You can use an Excel feature called Pointing
excel-data-and-formula

Pointing allows you to click with your mouse on the cell that contains the data or approach a cell reference using the keyboard ARROW keys containing your data to add. This will add cell reference to the formula.

After typing an equal sign in cell C3 in step 2:

  1. Click on cell $A1$ with the mouse pointer to enter the cell reference into the formula
  2. Type a plus (+) sign. You can also use other operators such as for multiplication use you have to use the * symbol, for division/symbol, and for subtraction use $–$, etc.
  3. Click on cell $A2$ with the mouse pointer to enter the cell reference into the formula
  4. Press the ENTER key on the keyboard

The answer 5 should appear in cell $C1$.

Note if you have more than one row or column of data then you need to perform calculations on each row or column cell. It is often possible to copy the first formula to other cells. The easiest way is to copy formulas with the file handle.

See also Creating Formulas in Microsoft Excel

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Stationary Stochastic Process (2016)

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Stationary Stochastic Process

A stationary stochastic process is said to be stationary if its mean and variance are constant over time and the value of the covariance between the two time periods depends only on a distance or gap or lag between the two time periods and not the actual time at which the covariance is computed. Such a stochastic process is also known as weak stationary, covariance stationary, second-order stationary, or wide-sense stochastic process.

In other words, a sequence of random variables {$y_t$} is covariance stationary if there is no trend, and if the covariance does not change over time.

Stationary Stochastic Process

Strictly Stationary (Covariance Stationary)

A time series is strictly stationary if all the moments of its probability distribution are invariance over time but not for the first two (mean and variance).

Let $y_t$ be a stochastic time series with

$E(y_t) = \mu $    $\Rightarrow$ Mean
$V(y_t) = E(y_t -\mu)^2=\sigma^2 $  $\Rightarrow$ Variance
$\gamma_k = E[(y_t-\mu)(y_{t+k}-\mu)]$  $\Rightarrow$ Covariance = $Cov(y_t, y_{t-k})$

$\gamma_k$ is covariance or autocovariance at lag $k$.

If $k=0$ then $Var(y_t)=\sigma^2$ i.e. $Cov(y_t)=Var(y_t)=\sigma^2$

If $k=1$ then we have covariance between two adjacent values of $y$.

If $y_t$ is to be stationary, the mean, variance, and autocovariance of $y_{t+m}$ (shift or origin of $y=m$) must be the same as those of $y_t$. OR

If a time series is stationary, its mean, variance, and autocovariance remain the same no matter at what point we measure them, i.e., they are time-invariant.

Non-Stationary Time Series

A time series having a time-varying mean or a time-varying variance or both is called a non-stationary time series.

Purely Random/ White Noise Process

A stochastic process having zero mean and constant variance ($\sigma^2$) and serially uncorrelated is called a purely random/ white noise process.

If it is independent also then such a process is called strictly white noise.

White noise denoted by $\mu_t$ as $\mu_t \sim N(0, \sigma^2)$ i.e. $\mu_t$ is independently and identically distributed as a normal distribution with zero mean and constant variance.

A stationary time series is important because if a time series is non-stationary, we can study its behavior only for the time period under consideration. Each set of time series data will, therefore, be for a particular episode. As a consequence, it is not possible to generalize it to other time periods. Therefore, for forecasting, such (non-stochastic) time series may be of little practical value. Our interest is in stationary time series.

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A Comprehensive Guide to Binomial Distribution (2016)

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In this post, we will learn about Binomial Distribution and its basics.

A statistical experiment having successive independent trials having two possible outcomes (such as success and failure; true and false; yes and no; right and wrong etc.) and probability of success is equal for each trial, while this kind of experiment is repeated a fixed number of times (say $n$ times) is called Binomial Experiment, Each trial of this Binomial experiment is known as Bernoulli trial (a trial which is a single performance of an experiment), for example.

Properties of the Binomial Experiment

  1. Each trial of the Binomial Experiment can be classified as a success or failure.
  2. The probability of success for each trial of the experiment is equal.
  3. Successive trials are independent, that is, the occurrence of one outcome in an experiment does not affect the occurrence of the other.
  4. The experiment is repeated a fixed number of times.

Binomial Distribution

Let $X$ be a discrete random variable, which denotes the number of successes of a Binomial Experiment (we call this binomial random variable). The random variable assumes isolated values as $X=0,1,2,\cdots,n$. The probability distribution of the binomial random variables is termed binomial distribution. It is a discrete probability distribution.

Binomial Probability Mass Function

The probability function of the binomial distribution is also called binomial probability mass function and can be denoted by $b(x, n, p)$, that is, a binomial distribution of random variable $X$ with $n$ (given number of trials) and $p$ (probability of success) as parameters. If $p$ is the probability of success (alternatively $q=1-p$ is probability of failure such that $p+q=1$) then probability of exactly $x$ success can be found from the following formula,

\begin{align}
b(x, n, p) &= P(X=x)\\
&=\binom{n}{x} p^x q^{n-x}, \quad x=0,1,2, \cdots, n
\end{align}

where $p$ is the probability of success of a single trial, $q$ is the probability of failure and $n$ is the number of independent trials.

The formula gives the probability for each possible combination of $n$ and $p$ of a binomial random variable $X$. Note that it does not give $P(X <0)$ and $P(X>n)$. The binomial distribution is suitable when $n$ is small and is applied when sampling done is with replacement.

\[b(x, n, p) = \binom{n}{x} p^x q^{n-x}, \quad x=0,1,2,\cdots,n,\]

is called Binomial distribution because its successive terms are the same as that of binomial expansion of

Binomial Distribution

\begin{align}
(q+p)^n=\binom{0}{0} p^0 q^{n-0}+\binom{n}{1} p^1 q^{n-1}+\cdots+\binom{n}{n-1} p^n q^{n-(n-1)}+\binom{n}{n} p^n q^{n-n}
\end{align}

$\binom{n}{0}, \binom{n}{1}, \binom{n}{2},\cdots, \binom{n}{n-1}, \binom{n}{n}$ are called Binomial coefficients.

Note that it is necessary to describe the limit of the random variable otherwise, it will be only the mathematical equation, not the probability distribution.

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