Avoiding False Cointegration from Misaligned Hedge-Ratio Multiplication

# Why does this Co-integrated basket look too good to be true?


# Why does this Co-integrated basket look too good to be true?












You need quantmod & tseries in R to run this:

```
library(quantmod)
library(tseries)

pairs <- c(
    "EUR/USD",
    "GBP/USD",
    "AUD/USD",
    "USD/CAD",
    "USD/CHF",
    "NZD/USD"
    )

name <- function(n) {
    gsub("/","",n,fixed=TRUE)
}

getFrame <- function(p) {
    result <- NULL
    as.data.frame(lapply(p, function(x) {
        if(!exists(name(x))) {
            getSymbols(x, src="oanda") 
        }
        if(is.null(result)) {
            result <- get(name(x))
        } else {
            result <- merge(result, get(name(x))) 
        }
    }))
}

isStationary <- function(frame) {
    model <- lm(frame[,1] ~ as.matrix(frame[,-1]) + 0)
    spread <- frame[,1] - rowSums(coef(model) * frame[, -1])
    results <- adf.test(spread, alternative="stationary", k=10)
    if(results$p.value < 0.05) {
			coefficients <- coef(model)
			names(coefficients) <- gsub("as.matrix.frame.......", "", names(coefficients))
			plot(spread[1:100], type = "b")
			cat("Minimum spread: ", min(spread), "\n")
			cat("Maximum spread: ", max(spread), "\n")
			cat("P-Value: ", results$p.value, "\n")
        cat("Coeficients: \n")
        print(coefficients)
    }
}

frame <- getFrame(pairs)
isStationary(frame)
```

I get FX daily data from Oanda, do a simple linear regression to find the hedging ratios, and then use the Augmented DF test to test for the P-value of mean reversion in the spread.

When I run it I get this:

```
Minimum spread:  -1.894506 
Maximum spread:  2.176735 
P-Value:  0.03781909 
Coeficients: 
        GBP.USD     AUD.USD     USD.CAD     USD.CHF     NZD.USD 
 0.59862816  0.48810239 -0.12900886  0.04337268  0.02713479
```

EUR.USD coefficient is 1.

When I plot the spread the first 100 days look like this:

Surely something must be wrong. The holy grail shouldn't be so easy to find.

Can someone help me find what is wrong?

I tried backtesting on Dukascopy with the above coefficients as lot sizes of a basket, but I run into loses. And the spread has a different order of magnitude in dukascopy. Why is that?

## Answer by Sergey (score 9, accepted)

https://quant.stackexchange.com/a/1709

The main problem in your code is this line:

`rowSums(coef(model) * frame[, -1])`

I'm not sure exactly what is does, perhaps some matrix multiplication, but definitely not what you expect it to do. Try to replace it with manual multiplication

`spread <- frame[,1] - (coef(model)[1]*frame[,2] + coef(model)[2]*frame[,3] + coef(model)[3]*frame[,4] + coef(model)[4]*frame[,5] + coef(model)[5]*frame[,6])`

And holy grail will disappear

I can see a couple of other errors as well:

- You cant sell on ASK price. With getSymbols.oanda you always get ASK

- You'd better separate testing and training data sets

## Answer by Joshua Ulrich (score 3)

https://quant.stackexchange.com/a/1715

@Sergey correctly identified the problem. The explanation is that `coef(model)` is a vector, `frame` is a data.frame, and element-by-element multiplication takes place in column-major order. The shorter vector (`coef(model)`) is recycled along the longer vector (each column in `frame`). For example:

```
frame <- data.frame(V1=1:5)
frame$V2 <- 2
frame$V3 <- 5
coef.model <- 1:3
frame * coef.model
#   V1 V2 V3
# 1  1  6 10
# 2  4  2 15
# 3  9  4  5
# 4  4  6 10
# 5 10  2 15
```

What you intended was something like this:

```
sweep(frame,2,coef.model,"*")
#   V1 V2 V3
# 1  1  4 15
# 2  2  4 15
# 3  3  4 15
# 4  4  4 15
# 5  5  4 15
```