# Analysis of Time Series data

The below graph is a scatterplot of daily stock price. My aim is to predict future stock price of the company.

From the scatterplot it seems that it is a multiplicative model, so I tried to "decompose" it in R. However it says that "time series has no or less than 2 periods". I also obtained a periodogram, which has only one peak at frequency close to 0.

However, my teacher told me that this time series cannot have a trend therefore to eliminate the seasonality I have to consider its period as 7 and then eliminate it choosing an appropriate model. Can anyone tell me what could be an appropriate model along with a proper justification? Also is it true that the series cannot have a trend?

Speaking as someone from a finance background, the `usual' model for a stock price process is

$$\frac{dS}{S}=r dt + \sigma dW_t$$

i.e. we assume the returns (not the absolute price changes, i.e. dS/S is approximately the daily percentage change in price) as having a 'drift' equal to the risk free rate (the interest rate r) and a random shock $$dW_t$$ with standard deviation $$\sigma$$

So it's basically the opposite of what you've been asked, I would expect a trend (but it's very hard to estimate with confidence) and I would not expect any seasonality, you could fit an arithmetic model by taking the daily differences though.

Have you tried taking the first difference? This amounts to taking the first derivative, and is generally a good way to de-trend a time series.

However, if you want to use seasonality, fit a regression model of form $$X_t = X_{t-k} + \epsilon$$ where $$k$$ is the number of time periods between seasons. For example, if you have monthly observations, using $$k=12$$ might make sense, as this removes the annual seasonality.

• Isn't the above formula valid if time series is additive? Also how did you know that the given time series is additive? – Jor_El Feb 21 '19 at 9:29
• The above construct only helps remove the seasonality from $k$ time periods ago. If the time series does not have seasonality, the regression model should have very small coefficients. You can also use regularization to determine if the coefficients should be non-zero. – David Atlas Feb 21 '19 at 12:58
• What model(additive or multiplicative) do you think the above time series follows?Also, how would I check whether seasonality is present or not? – Jor_El Feb 21 '19 at 20:53
• See this link for more information on how to tell if a time series is additive or multiplicative:r-bloggers.com/is-my-time-series-additive-or-multiplicative My answer above shows how to test for seasonality. Simply use the p-values of the regression model to inform if significant seasonality is present. – David Atlas Feb 21 '19 at 21:17