Pearson correlation changes after transforming variables (time series, ARIMA) - what is wrong?

I have four variables (Y, X1, X2, X3). The original Y and original X1 both have positive trends and have a Pearson Correlation 0.845 (significantly positive correlation). Based on theories, the hypothesized model is Y= X1 + X2 + X3. Then I first log the four variables, and run an ARIMA (1,1,1) on the four variables, using the first difference.In the output of ARIMA (1,1,1), log Y= -.052 log X1+ ....... The relationship between Y and X1 became "negative (-.052)".

(1) Is this normal or ok? What is wrong? How to fix this? (The initial correlation between Y and X1 was positive 0.845. Now it becomes -.052.)

(2)Can I interpret that the relationship between X1 and Y is: One percent increase in X is associated with .052 percent "decrease" in Y. Is this the correct way to interpret ARIMA output?

Thank you.

• Nothing wrong. You had one correlation for $(x,y)$ and a different correlation for $(f(x),g(y))$, where $f(\cdot)$ and $g(\cdot)$ are some functions. Why should the two correlations be the same or even similar? Also, it is unclear to me what exactly the variables are in the second correlation. I understand that $g(y)=\log(y)$. But for $f(x)$, is it the fitted values from the ARIMA model, or the residuals from the ARIMA model, or yet something else? – Richard Hardy Sep 6 '17 at 9:39
• Why would you think the correlation would be unchanged? If you want a correlation measure to be invariant to monotonic transformations, look at a Kendall or a Spearman correlation. – Glen_b Sep 6 '17 at 10:22
• How do you have a relationship between Y and X1 in an ARIMA model? ARIMA models Y on itself, not on some other variable. What are you actually doing? Also please put such information in your question and use comments to briefly mention changes – Glen_b Sep 6 '17 at 12:04
• Multivariate ARIMA? You really need to clearly explain that IN THE QUESTION. Why would you leave this information out? How are we to guess? Please make the models and steps clear – Glen_b Sep 6 '17 at 12:10
• Don't post a new one. Please edit your question. There's an "edit" link under your post. – Glen_b Sep 6 '17 at 12:17